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147-578: The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. The Schrödinger equation deals with state vectors , which can only describe pure quantum states and are thus less general than density matrices , which can describe mixed states as well. In

294-657: A 0 ( 2 r n a 0 ) ℓ L n − ℓ − 1 2 ℓ + 1 ( 2 r n a 0 ) ⋅ Y ℓ m ( θ , φ ) {\displaystyle \psi _{n\ell m}(r,\theta ,\varphi )={\sqrt {\left({\frac {2}{na_{0}}}\right)^{3}{\frac {(n-\ell -1)!}{2n[(n+\ell )!]}}}}e^{-r/na_{0}}\left({\frac {2r}{na_{0}}}\right)^{\ell }L_{n-\ell -1}^{2\ell +1}\left({\frac {2r}{na_{0}}}\right)\cdot Y_{\ell }^{m}(\theta ,\varphi )} where It

441-447: A Hamiltonian dynamical system with canonical coordinates q i {\displaystyle q_{i}} and conjugate momenta p i {\displaystyle p_{i}} , where i = 1 , … , n {\displaystyle i=1,\dots ,n} . Then the phase space distribution ρ ( p , q ) {\displaystyle \rho (p,q)} determines

588-922: A Hermitian matrix . Separation of variables can also be a useful method for the time-independent Schrödinger equation. For example, depending on the symmetry of the problem, the Cartesian axes might be separated, ψ ( r ) = ψ x ( x ) ψ y ( y ) ψ z ( z ) , {\displaystyle \psi (\mathbf {r} )=\psi _{x}(x)\psi _{y}(y)\psi _{z}(z),} or radial and angular coordinates might be separated: ψ ( r ) = ψ r ( r ) ψ θ ( θ ) ψ ϕ ( ϕ ) . {\displaystyle \psi (\mathbf {r} )=\psi _{r}(r)\psi _{\theta }(\theta )\psi _{\phi }(\phi ).} The particle in

735-585: A single formulation that simplifies to the Schrödinger equation in the non-relativistic limit. This is the Dirac equation , which contains a single derivative in both space and time. The second-derivative PDE of the Klein-Gordon equation led to a problem with probability density even though it was a relativistic wave equation . The probability density could be negative, which is physically unviable. This

882-443: A Lindbladian for various times are collectively referred to as a quantum dynamical semigroup —a family of quantum dynamical maps ϕ t {\displaystyle \phi _{t}} on the space of density matrices indexed by a single time parameter t ≥ 0 {\displaystyle t\geq 0} that obey the semigroup property The Lindblad equation can be obtained by which, by

1029-1002: A basis for the space of operators. The general form is not in fact more general, and can be reduced to the special form. Since the matrix h is positive semidefinite, it can be diagonalized with a unitary transformation u : where the eigenvalues γ i are non-negative. If we define another orthonormal operator basis This reduces the master equation to the same form as before: ρ ˙ = − i ℏ [ H , ρ ] + ∑ i γ i ( L i ρ L i † − 1 2 { L i † L i , ρ } ) {\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)} The maps generated by

1176-435: A classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by commutators . In this case, the resulting equation is where ρ is the density matrix . When applied to the expectation value of an observable , the corresponding equation is given by Ehrenfest's theorem , and takes the form where A {\displaystyle A}

1323-415: A constraint on the energy levels, yielding E n = ℏ 2 π 2 n 2 2 m L 2 = n 2 h 2 8 m L 2 . {\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.} A finite potential well

1470-414: A given system, we can consider the phase space ( q μ , p μ ) {\displaystyle (q^{\mu },p_{\mu })} of a particular Hamiltonian H {\displaystyle H} as a manifold ( M , ω ) {\displaystyle (M,\omega )} endowed with a symplectic 2-form The volume form of our manifold

1617-865: A hydrogen atom can be solved by separation of variables. In this case, spherical polar coordinates are the most convenient. Thus, ψ ( r , θ , φ ) = R ( r ) Y ℓ m ( θ , φ ) = R ( r ) Θ ( θ ) Φ ( φ ) , {\displaystyle \psi (r,\theta ,\varphi )=R(r)Y_{\ell }^{m}(\theta ,\varphi )=R(r)\Theta (\theta )\Phi (\varphi ),} where R are radial functions and Y l m ( θ , φ ) {\displaystyle Y_{l}^{m}(\theta ,\varphi )} are spherical harmonics of degree ℓ {\displaystyle \ell } and order m {\displaystyle m} . This

SECTION 10

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1764-411: A mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function , the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave . The equation predicted bound states of

1911-676: A one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy inside a certain region and infinite potential energy outside . For the one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written − ℏ 2 2 m d 2 ψ d x 2 = E ψ . {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .} With

2058-556: A perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise a position eigenstate would be a Dirac delta distribution , not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes regard these eigenstates, composed of elements outside the Hilbert space, as " generalized eigenvectors ". These are used for calculational convenience and do not represent physical states. Thus,

2205-647: A position-space wave function Ψ ( x , t ) {\displaystyle \Psi (x,t)} as used above can be written as the inner product of a time-dependent state vector | Ψ ( t ) ⟩ {\displaystyle |\Psi (t)\rangle } with unphysical but convenient "position eigenstates" | x ⟩ {\displaystyle |x\rangle } : Ψ ( x , t ) = ⟨ x | Ψ ( t ) ⟩ . {\displaystyle \Psi (x,t)=\langle x|\Psi (t)\rangle .} The form of

2352-484: A smooth measure (locally, this measure is the 6 n -dimensional Lebesgue measure ). The theorem says this smooth measure is invariant under the Hamiltonian flow . More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow . The Hamiltonian case then becomes a corollary. We can also formulate Liouville's Theorem in terms of symplectic geometry . For

2499-703: A system of multiple particles, each one will have a phase-space trajectory that traces out an ellipse corresponding to the particle's energy. The frequency at which the ellipse is traced is given by the ω {\displaystyle \omega } in the Hamiltonian, independent of any differences in energy. As a result, a region of phase space will simply rotate about the point ( q , p ) = ( 0 , 0 ) {\displaystyle (\mathbf {q} ,\mathbf {p} )=(0,0)} with frequency dependent on ω {\displaystyle \omega } . This can be seen in

2646-731: A three-dimensional position vector and p {\displaystyle \mathbf {p} } for a three-dimensional momentum vector, the position-space Schrödinger equation is i ℏ ∂ ∂ t Ψ ( r , t ) = − ℏ 2 2 m ∇ 2 Ψ ( r , t ) + V ( r ) Ψ ( r , t ) . {\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t)+V(\mathbf {r} )\Psi (\mathbf {r} ,t).} The momentum-space counterpart involves

2793-420: A very similar procedure to the undamped harmonic oscillator case, and we arrive again at Plugging in our modified Hamilton's equations, we find Calculating our new infinitesimal phase space volume, and keeping only first order in δ t {\displaystyle \delta t} we find the following result: We have found that the infinitesimal phase-space volume is no longer constant, and thus

2940-446: Is unital , i.e. it preserves the identity operator. The Lindblad master equation describes the evolution of various types of open quantum systems, e.g. a system weakly coupled to a Markovian reservoir. Note that the H appearing in the equation is not necessarily equal to the bare system Hamiltonian, but may also incorporate effective unitary dynamics arising from the system-environment interaction. A heuristic derivation, e.g. , in

3087-631: Is a conserved current . Notice that the difference between this and Liouville's equation are the terms where H {\displaystyle H} is the Hamiltonian, and where the derivatives ∂ q ˙ i / ∂ q i {\displaystyle \partial {\dot {q}}_{i}/\partial q_{i}} and ∂ p ˙ i / ∂ p i {\displaystyle \partial {\dot {p}}_{i}/\partial p_{i}} have been evaluated using Hamilton's equations of motion. That is, viewing

SECTION 20

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3234-584: Is a linear differential equation , meaning that if two state vectors | ψ 1 ⟩ {\displaystyle |\psi _{1}\rangle } and | ψ 2 ⟩ {\displaystyle |\psi _{2}\rangle } are solutions, then so is any linear combination | ψ ⟩ = a | ψ 1 ⟩ + b | ψ 2 ⟩ {\displaystyle |\psi \rangle =a|\psi _{1}\rangle +b|\psi _{2}\rangle } of

3381-697: Is a continuous family of unitary operators parameterized by t {\displaystyle t} . Without loss of generality , the parameterization can be chosen so that U ^ ( 0 ) {\displaystyle {\hat {U}}(0)} is the identity operator and that U ^ ( t / N ) N = U ^ ( t ) {\displaystyle {\hat {U}}(t/N)^{N}={\hat {U}}(t)} for any N > 0 {\displaystyle N>0} . Then U ^ ( t ) {\displaystyle {\hat {U}}(t)} depends upon

3528-412: Is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and τ ( t ) {\displaystyle \tau (t)} is a function of time only. Substituting this expression for Ψ {\displaystyle \Psi } into the time dependent left hand side shows that τ ( t ) {\displaystyle \tau (t)}

3675-406: Is a phase factor: Ψ ( r , t ) = ψ ( r ) e − i E t / ℏ . {\displaystyle \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )e^{-i{Et/\hbar }}.} A solution of this type is called stationary, since the only time dependence is a phase factor that cancels when the probability density

3822-510: Is a real function which represents the complex phase of the wavefunction, then the probability flux is calculated as: j = ρ ∇ S m {\displaystyle \mathbf {j} ={\frac {\rho \nabla S}{m}}} Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. Although the ∇ S m {\textstyle {\frac {\nabla S}{m}}} term appears to play

3969-413: Is a wave function, a function that assigns a complex number to each point x {\displaystyle x} at each time t {\displaystyle t} . The parameter m {\displaystyle m} is the mass of the particle, and V ( x , t ) {\displaystyle V(x,t)} is the potential that represents the environment in which

4116-685: Is an observable. Note the sign difference, which follows from the assumption that the operator is stationary and the state is time-dependent. In the phase-space formulation of quantum mechanics, substituting the Moyal brackets for Poisson brackets in the phase-space analog of the von Neumann equation results in compressibility of the probability fluid , and thus violations of Liouville's theorem incompressibility. This, then, leads to concomitant difficulties in defining meaningful quantum trajectories. Consider an N {\displaystyle N} -particle system in three dimensions, and focus on only

4263-417: Is available in the mathematical setting of symplectic geometry . Liouville's theorem ignores the possibility of chemical reactions , where the total number of particles may change over time, or where energy may be transferred to internal degrees of freedom . There are extensions of Liouville's theorem to cover these various generalized settings, including stochastic systems. The Liouville equation describes

4410-505: Is calculated via the Born rule. The spatial part of the full wave function solves: ∇ 2 ψ ( r ) + 2 m ℏ 2 [ E − V ( r ) ] ψ ( r ) = 0. {\displaystyle \nabla ^{2}\psi (\mathbf {r} )+{\frac {2m}{\hbar ^{2}}}\left[E-V(\mathbf {r} )\right]\psi (\mathbf {r} )=0.} where

4557-797: Is called the ground state , its energy is called the zero-point energy , and the wave function is a Gaussian . The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized. The Schrödinger equation for the electron in a hydrogen atom (or a hydrogen-like atom) is E ψ = − ℏ 2 2 μ ∇ 2 ψ − q 2 4 π ε 0 r ψ {\displaystyle E\psi =-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi -{\frac {q^{2}}{4\pi \varepsilon _{0}r}}\psi } where q {\displaystyle q}

Lindbladian - Misplaced Pages Continue

4704-524: Is closely related to the Hamilton–Jacobi equation (HJE) − ∂ ∂ t S ( q i , t ) = H ( q i , ∂ S ∂ q i , t ) {\displaystyle -{\frac {\partial }{\partial t}}S(q_{i},t)=H\left(q_{i},{\frac {\partial S}{\partial q_{i}}},t\right)} where S {\displaystyle S}

4851-575: Is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of square-integrable functions L 2 {\displaystyle L^{2}} , while the Hilbert space for the spin of a single proton is the two-dimensional complex vector space C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables , which are self-adjoint operators acting on

4998-759: Is explicitly time dependent. Also, according to the interaction picture, χ ~ = U B S ( t , t 0 ) χ U B S † ( t , t 0 ) {\displaystyle {\tilde {\chi }}=U_{BS}(t,t_{0})\chi U_{BS}^{\dagger }(t,t_{0})} , where U B S = U 0 † U ( t , t 0 ) {\displaystyle U_{BS}=U_{0}^{\dagger }U(t,t_{0})} . This equation can be integrated directly to give This implicit equation for χ ~ {\displaystyle {\tilde {\chi }}} can be substituted back into

5145-402: Is given by the ellipse of constant H {\displaystyle H} . Explicitly, one can solve Hamilton's equations for the system and find where Q i {\displaystyle Q_{i}} and P i {\displaystyle P_{i}} denote the initial position and momentum of the i {\displaystyle i} -th particle. For

5292-426: Is given in the previous example. This time, we add the condition that each particle experiences a frictional force − γ p i {\displaystyle -\gamma p_{i}} , where γ {\displaystyle \gamma } is a positive constant dictating the amount of friction. As this is a non-conservative force , we need to extend Hamilton's equations as Unlike

5439-882: Is illustrated by the position-space and momentum-space Schrödinger equations for a nonrelativistic, spinless particle. The Hilbert space for such a particle is the space of complex square-integrable functions on three-dimensional Euclidean space, and its Hamiltonian is the sum of a kinetic-energy term that is quadratic in the momentum operator and a potential-energy term: i ℏ d d t | Ψ ( t ) ⟩ = ( 1 2 m p ^ 2 + V ^ ) | Ψ ( t ) ⟩ . {\displaystyle i\hbar {\frac {d}{dt}}|\Psi (t)\rangle =\left({\frac {1}{2m}}{\hat {p}}^{2}+{\hat {V}}\right)|\Psi (t)\rangle .} Writing r {\displaystyle \mathbf {r} } for

5586-443: Is impossible to solve analytically except in very particular cases. What's more, under certain approximations, the bath degrees of freedom need not be considered, and an effective master equation can be derived in terms of the system density matrix, ρ = tr B ⁡ χ {\displaystyle \rho =\operatorname {tr} _{B}\chi } . The problem can be analyzed more easily by moving into

5733-453: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ | P λ | ψ ⟩ {\displaystyle \langle \psi |P_{\lambda }|\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. A momentum eigenstate would be

5880-915: Is known as the time-evolution operator, and it is unitary : it preserves the inner product between vectors in the Hilbert space. Unitarity is a general feature of time evolution under the Schrödinger equation. If the initial state is | Ψ ( 0 ) ⟩ {\displaystyle |\Psi (0)\rangle } , then the state at a later time t {\displaystyle t} will be given by | Ψ ( t ) ⟩ = U ^ ( t ) | Ψ ( 0 ) ⟩ {\displaystyle |\Psi (t)\rangle ={\hat {U}}(t)|\Psi (0)\rangle } for some unitary operator U ^ ( t ) {\displaystyle {\hat {U}}(t)} . Conversely, suppose that U ^ ( t ) {\displaystyle {\hat {U}}(t)}

6027-753: Is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories. For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point x 0 {\displaystyle x_{0}} , then V ′ ( ⟨ X ⟩ ) {\displaystyle V'\left(\left\langle X\right\rangle \right)} and ⟨ V ′ ( X ) ⟩ {\displaystyle \left\langle V'(X)\right\rangle } will be almost

Lindbladian - Misplaced Pages Continue

6174-594: Is now explicit in the system degrees of freedom, but is very difficult to solve. A final assumption is the Born-Markov approximation that the time derivative of the density matrix depends only on its current state, and not on its past. This assumption is valid under fast bath dynamics, wherein correlations within the bath are lost extremely quickly, and amounts to replacing ρ ( t ′ ) → ρ ( t ) {\displaystyle \rho (t')\rightarrow \rho (t)} on

6321-458: Is often restated in terms of the Poisson bracket as or, in terms of the linear Liouville operator or Liouvillian , as In ergodic theory and dynamical systems , motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem. In Hamiltonian mechanics , the phase space is a smooth manifold that comes naturally equipped with

6468-400: Is preserved, not only its top exterior power. That is, Liouville's Theorem also gives The analog of Liouville equation in quantum mechanics describes the time evolution of a mixed state . Canonical quantization yields a quantum-mechanical version of this theorem, the von Neumann equation . This procedure, often used to devise quantum analogues of classical systems, involves describing

6615-407: Is sometimes called "wave mechanics". The Klein-Gordon equation is a wave equation which is the relativistic version of the Schrödinger equation. The Schrödinger equation is nonrelativistic because it contains a first derivative in time and a second derivative in space, and therefore space & time are not on equal footing. Paul Dirac incorporated special relativity and quantum mechanics into

6762-613: Is taken to define a probability density function . For example, given a wave function in position space Ψ ( x , t ) {\displaystyle \Psi (x,t)} as above, we have Pr ( x , t ) = | Ψ ( x , t ) | 2 . {\displaystyle \Pr(x,t)=|\Psi (x,t)|^{2}.} The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves , called stationary states . These states are particularly important as their individual study later simplifies

6909-464: Is the anticommutator . H {\displaystyle H} is the system Hamiltonian , describing the unitary aspects of the dynamics. { L i } i {\displaystyle \{L_{i}\}_{i}} are a set of jump operators , describing the dissipative part of the dynamics. The shape of the jump operators describes how the environment acts on the system, and must either be determined from microscopic models of

7056-488: Is the permittivity of free space and μ = m q m p m q + m p {\displaystyle \mu ={\frac {m_{q}m_{p}}{m_{q}+m_{p}}}} is the 2-body reduced mass of the hydrogen nucleus (just a proton ) of mass m p {\displaystyle m_{p}} and the electron of mass m q {\displaystyle m_{q}} . The negative sign arises in

7203-548: Is the probability current or probability flux (flow per unit area). If the wavefunction is represented as ψ ( x , t ) = ρ ( x , t ) exp ⁡ ( i S ( x , t ) ℏ ) , {\textstyle \psi ({\bf {x}},t)={\sqrt {\rho ({\bf {x}},t)}}\exp \left({\frac {iS({\bf {x}},t)}{\hbar }}\right),} where S ( x , t ) {\displaystyle S(\mathbf {x} ,t)}

7350-749: Is the basis of energy eigenstates, which are solutions of the time-independent Schrödinger equation. In this basis, a time-dependent state vector | Ψ ( t ) ⟩ {\displaystyle |\Psi (t)\rangle } can be written as the linear combination | Ψ ( t ) ⟩ = ∑ n A n e − i E n t / ℏ | ψ E n ⟩ , {\displaystyle |\Psi (t)\rangle =\sum _{n}A_{n}e^{{-iE_{n}t}/\hbar }|\psi _{E_{n}}\rangle ,} where A n {\displaystyle A_{n}} are complex numbers and

7497-1028: Is the classical action and H {\displaystyle H} is the Hamiltonian function (not operator). Here the generalized coordinates q i {\displaystyle q_{i}} for i = 1 , 2 , 3 {\displaystyle i=1,2,3} (used in the context of the HJE) can be set to the position in Cartesian coordinates as r = ( q 1 , q 2 , q 3 ) = ( x , y , z ) {\displaystyle \mathbf {r} =(q_{1},q_{2},q_{3})=(x,y,z)} . Substituting Ψ = ρ ( r , t ) e i S ( r , t ) / ℏ {\displaystyle \Psi ={\sqrt {\rho (\mathbf {r} ,t)}}e^{iS(\mathbf {r} ,t)/\hbar }} where ρ {\displaystyle \rho }

SECTION 50

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7644-405: Is the density operator of the bath initially. Tracing over the bath degrees of freedom, tr R ⁡ χ ~ = ρ ~ {\displaystyle \operatorname {tr} _{R}{\tilde {\chi }}={\tilde {\rho }}} , of the aforementioned differo-integral equation yields This equation is exact for the time dynamics of

7791-1243: Is the displacement and ω {\displaystyle \omega } the angular frequency. Furthermore, it can be used to describe approximately a wide variety of other systems, including vibrating atoms, molecules , and atoms or ions in lattices, and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics. The solutions in position space are ψ n ( x ) = 1 2 n n !   ( m ω π ℏ ) 1 / 4   e − m ω x 2 2 ℏ   H n ( m ω ℏ x ) , {\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\ \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\ e^{-{\frac {m\omega x^{2}}{2\hbar }}}\ {\mathcal {H}}_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),} where n ∈ { 0 , 1 , 2 , … } {\displaystyle n\in \{0,1,2,\ldots \}} , and

7938-469: Is the electron charge, r {\displaystyle \mathbf {r} } is the position of the electron relative to the nucleus, r = | r | {\displaystyle r=|\mathbf {r} |} is the magnitude of the relative position, the potential term is due to the Coulomb interaction , wherein ε 0 {\displaystyle \varepsilon _{0}}

8085-503: Is the energy of the system. This is only used when the Hamiltonian itself is not dependent on time explicitly. However, even in this case the total wave function is dependent on time as explained in the section on linearity below. In the language of linear algebra , this equation is an eigenvalue equation . Therefore, the wave function is an eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s) E {\displaystyle E} . The Schrödinger equation

8232-408: Is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside

8379-521: Is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximate methods. The family of solutions are: ψ n ℓ m ( r , θ , φ ) = ( 2 n a 0 ) 3 ( n − ℓ − 1 ) ! 2 n [ ( n + ℓ ) ! ] e − r / n

8526-401: Is the probability density, into the Schrödinger equation and then taking the limit ℏ → 0 {\displaystyle \hbar \to 0} in the resulting equation yield the Hamilton–Jacobi equation . Wave functions are not always the most convenient way to describe quantum systems and their behavior. When the preparation of a system is only imperfectly known, or when

8673-486: Is the quantum analog of the classical Liouville equation . The entire equation can be written in superoperator form : ρ ˙ = L ( ρ ) {\displaystyle {\dot {\rho }}={\mathcal {L}}(\rho )} which resembles the classical Liouville equation ρ ˙ = { H , ρ } {\displaystyle {\dot {\rho }}=\{H,\rho \}} . For this reason,

8820-551: Is the top exterior power of the symplectic 2-form, and is just another representation of the measure on the phase space described above. On our phase space symplectic manifold we can define a Hamiltonian vector field generated by a function f ( q , p ) {\displaystyle f(q,p)} as Specifically, when the generating function is the Hamiltonian itself, f ( q , p ) = H {\displaystyle f(q,p)=H} , we get where we utilized Hamilton's equations of motion and

8967-533: Is the total unitary operator of the entire system. It is straightforward to confirm that the Liouville equation becomes where the Hamiltonian H ~ B S = e i ( H S + H B ) t H B S e − i ( H S + H B ) t {\displaystyle {\tilde {H}}_{BS}=e^{i(H_{S}+H_{B})t}H_{BS}e^{-i(H_{S}+H_{B})t}}

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9114-484: Is time, | Ψ ( t ) ⟩ {\displaystyle \vert \Psi (t)\rangle } is the state vector of the quantum system ( Ψ {\displaystyle \Psi } being the Greek letter psi ), and H ^ {\displaystyle {\hat {H}}} is an observable, the Hamiltonian operator . The term "Schrödinger equation" can refer to both

9261-938: Is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem . For a one-dimensional quantum particle moving in a potential V {\displaystyle V} , the Ehrenfest theorem says m d d t ⟨ x ⟩ = ⟨ p ⟩ ; d d t ⟨ p ⟩ = − ⟨ V ′ ( X ) ⟩ . {\displaystyle m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .} Although

9408-435: Is typically not possible to solve the Schrödinger equation exactly for situations of physical interest. Accordingly, approximate solutions are obtained using techniques like variational methods and WKB approximation . It is also common to treat a problem of interest as a small modification to a problem that can be solved exactly, a method known as perturbation theory . One simple way to compare classical to quantum mechanics

9555-488: Is typically not the same as − ⟨ V ′ ( X ) ⟩ {\displaystyle -\left\langle V'(X)\right\rangle } . For a general V ′ {\displaystyle V'} , therefore, quantum mechanics can lead to predictions where expectation values do not mimic the classical behavior. In the case of the quantum harmonic oscillator, however, V ′ {\displaystyle V'}

9702-465: Is unitary only if, to first order, its derivative is Hermitian. The Schrödinger equation is often presented using quantities varying as functions of position, but as a vector-operator equation it has a valid representation in any arbitrary complete basis of kets in Hilbert space . As mentioned above, "bases" that lie outside the physical Hilbert space are also employed for calculational purposes. This

9849-465: The n -dimensional divergence theorem . This proof is based on the fact that the evolution of ρ {\displaystyle \rho } obeys an 2n -dimensional version of the continuity equation : That is, the 3-tuple ( ρ , ρ q ˙ i , ρ p ˙ i ) {\displaystyle (\rho ,\rho {\dot {q}}_{i},\rho {\dot {p}}_{i})}

9996-472: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ | ψ ⟩ | 2 {\displaystyle |\langle \lambda |\psi \rangle |^{2}} , where | λ ⟩ {\displaystyle |\lambda \rangle }

10143-2053: The Fourier transforms of the wave function and the potential: i ℏ ∂ ∂ t Ψ ~ ( p , t ) = p 2 2 m Ψ ~ ( p , t ) + ( 2 π ℏ ) − 3 / 2 ∫ d 3 p ′ V ~ ( p − p ′ ) Ψ ~ ( p ′ , t ) . {\displaystyle i\hbar {\frac {\partial }{\partial t}}{\tilde {\Psi }}(\mathbf {p} ,t)={\frac {\mathbf {p} ^{2}}{2m}}{\tilde {\Psi }}(\mathbf {p} ,t)+(2\pi \hbar )^{-3/2}\int d^{3}\mathbf {p} '\,{\tilde {V}}(\mathbf {p} -\mathbf {p} '){\tilde {\Psi }}(\mathbf {p} ',t).} The functions Ψ ( r , t ) {\displaystyle \Psi (\mathbf {r} ,t)} and Ψ ~ ( p , t ) {\displaystyle {\tilde {\Psi }}(\mathbf {p} ,t)} are derived from | Ψ ( t ) ⟩ {\displaystyle |\Psi (t)\rangle } by Ψ ( r , t ) = ⟨ r | Ψ ( t ) ⟩ , {\displaystyle \Psi (\mathbf {r} ,t)=\langle \mathbf {r} |\Psi (t)\rangle ,} Ψ ~ ( p , t ) = ⟨ p | Ψ ( t ) ⟩ , {\displaystyle {\tilde {\Psi }}(\mathbf {p} ,t)=\langle \mathbf {p} |\Psi (t)\rangle ,} where | r ⟩ {\displaystyle |\mathbf {r} \rangle } and | p ⟩ {\displaystyle |\mathbf {p} \rangle } do not belong to

10290-597: The Schrödinger picture can be equivalently described in the Heisenberg picture using the following (diagonalized) equation of motion for each quantum observable X : A similar equation describes the time evolution of the expectation values of observables, given by the Ehrenfest theorem . Corresponding to the trace-preserving property of the Schrödinger picture Lindblad equation, the Heisenberg picture equation

10437-527: The canonical commutation relation [ x ^ , p ^ ] = i ℏ . {\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar .} This implies that ⟨ x | p ^ | Ψ ⟩ = − i ℏ d d x Ψ ( x ) , {\displaystyle \langle x|{\hat {p}}|\Psi \rangle =-i\hbar {\frac {d}{dx}}\Psi (x),} so

10584-481: The French mathematician Joseph Liouville , is a key theorem in classical statistical and Hamiltonian mechanics . It asserts that the phase-space distribution function is constant along the trajectories of the system —that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as

10731-666: The Hamiltonian H ^ {\displaystyle {\hat {H}}} constant, the Schrödinger equation has the solution | Ψ ( t ) ⟩ = e − i H ^ t / ℏ | Ψ ( 0 ) ⟩ . {\displaystyle |\Psi (t)\rangle =e^{-i{\hat {H}}t/\hbar }|\Psi (0)\rangle .} The operator U ^ ( t ) = e − i H ^ t / ℏ {\displaystyle {\hat {U}}(t)=e^{-i{\hat {H}}t/\hbar }}

10878-708: The Hilbert space itself, but have well-defined inner products with all elements of that space. When restricted from three dimensions to one, the position-space equation is just the first form of the Schrödinger equation given above . The relation between position and momentum in quantum mechanics can be appreciated in a single dimension. In canonical quantization , the classical variables x {\displaystyle x} and p {\displaystyle p} are promoted to self-adjoint operators x ^ {\displaystyle {\hat {x}}} and p ^ {\displaystyle {\hat {p}}} that satisfy

11025-432: The Hilbert space. A wave function can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by

11172-776: The Hilbert space. These are the density-matrix representations of wave functions; in Dirac notation, they are written ρ ^ = | Ψ ⟩ ⟨ Ψ | . {\displaystyle {\hat {\rho }}=|\Psi \rangle \langle \Psi |.} The density-matrix analogue of the Schrödinger equation for wave functions is i ℏ ∂ ρ ^ ∂ t = [ H ^ , ρ ^ ] , {\displaystyle i\hbar {\frac {\partial {\hat {\rho }}}{\partial t}}=[{\hat {H}},{\hat {\rho }}],} where

11319-1471: The Klein Gordon equation, although a redefined inner product of a wavefunction can be time independent, the total volume integral of modulus square of the wavefunction need not be time independent. The continuity equation for probability in non relativistic quantum mechanics is stated as: ∂ ∂ t ρ ( r , t ) + ∇ ⋅ j = 0 , {\displaystyle {\frac {\partial }{\partial t}}\rho \left(\mathbf {r} ,t\right)+\nabla \cdot \mathbf {j} =0,} where j = 1 2 m ( Ψ ∗ p ^ Ψ − Ψ p ^ Ψ ∗ ) = − i ℏ 2 m ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) = ℏ m Im ⁡ ( ψ ∗ ∇ ψ ) {\displaystyle \mathbf {j} ={\frac {1}{2m}}\left(\Psi ^{*}{\hat {\mathbf {p} }}\Psi -\Psi {\hat {\mathbf {p} }}\Psi ^{*}\right)=-{\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {\hbar }{m}}\operatorname {Im} (\psi ^{*}\nabla \psi )}

11466-418: The Lindblad superoperator L is achieved. In the simplest case, there is just one jump operator F {\displaystyle F} and no unitary evolution. In this case, the Lindblad equation is This case is often used in quantum optics to model either absorption or emission of photons from a reservoir. To model both absorption and emission, one would need a jump operator for each. This leads to

11613-539: The Liouville equation to obtain an exact differo-integral equation We proceed with the derivation by assuming the interaction is initiated at t = 0 {\displaystyle t=0} , and at that time there are no correlations between the system and the bath. This implies that the initial condition is factorable as χ ( 0 ) = ρ ( 0 ) R 0 {\displaystyle \chi (0)=\rho (0)R_{0}} , where R 0 {\displaystyle R_{0}}

11760-525: The Schrödinger equation depends on the physical situation. The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time: i ℏ d d t | Ψ ( t ) ⟩ = H ^ | Ψ ( t ) ⟩ {\displaystyle i\hbar {\frac {d}{dt}}\vert \Psi (t)\rangle ={\hat {H}}\vert \Psi (t)\rangle } where t {\displaystyle t}

11907-513: The Schrödinger equation is often written for functions of momentum, as Bloch's theorem ensures the periodic crystal lattice potential couples Ψ ~ ( p ) {\displaystyle {\tilde {\Psi }}(p)} with Ψ ~ ( p + K ) {\displaystyle {\tilde {\Psi }}(p+K)} for only discrete reciprocal lattice vectors K {\displaystyle K} . This makes it convenient to solve

12054-417: The Schrödinger equation, write down the Hamiltonian for the system, accounting for the kinetic and potential energies of the particles constituting the system, then insert it into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system. In practice, the square of the absolute value of the wave function at each point

12201-433: The action of the momentum operator p ^ {\displaystyle {\hat {p}}} in the position-space representation is − i ℏ d d x {\textstyle -i\hbar {\frac {d}{dx}}} . Thus, p ^ 2 {\displaystyle {\hat {p}}^{2}} becomes a second derivative , and in three dimensions,

12348-417: The animation above. To see an example where Liouville's theorem does not apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping. Consider again the system of N {\displaystyle N} particles each in a 3 {\displaystyle 3} -dimensional isotropic harmonic potential, the Hamiltonian for which

12495-412: The atom in agreement with experimental observations. The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. Other formulations of quantum mechanics include matrix mechanics , introduced by Werner Heisenberg , and the path integral formulation , developed chiefly by Richard Feynman . When these approaches are compared, the use of the Schrödinger equation

12642-404: The bath degrees of freedom. By assuming rapid decay of these correlations (ideally ⟨ Γ i ( t ) Γ j ( t ′ ) ⟩ ∝ δ ( t − t ′ ) {\displaystyle \langle \Gamma _{i}(t)\Gamma _{j}(t')\rangle \propto \delta (t-t')} ), above form of

12789-997: The box determine the values of C , D , {\displaystyle C,D,} and k {\displaystyle k} at x = 0 {\displaystyle x=0} and x = L {\displaystyle x=L} where ψ {\displaystyle \psi } must be zero. Thus, at x = 0 {\displaystyle x=0} , ψ ( 0 ) = 0 = C sin ⁡ ( 0 ) + D cos ⁡ ( 0 ) = D {\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D} and D = 0 {\displaystyle D=0} . At x = L {\displaystyle x=L} , ψ ( L ) = 0 = C sin ⁡ ( k L ) , {\displaystyle \psi (L)=0=C\sin(kL),} in which C {\displaystyle C} cannot be zero as this would conflict with

12936-751: The brackets denote a commutator . This is variously known as the von Neumann equation, the Liouville–von Neumann equation, or just the Schrödinger equation for density matrices. If the Hamiltonian is time-independent, this equation can be easily solved to yield ρ ^ ( t ) = e − i H ^ t / ℏ ρ ^ ( 0 ) e i H ^ t / ℏ . {\displaystyle {\hat {\rho }}(t)=e^{-i{\hat {H}}t/\hbar }{\hat {\rho }}(0)e^{i{\hat {H}}t/\hbar }.} More generally, if

13083-446: The canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that there is no decay and phase coherence is maintained throughout the process, and is a consequence of the fact that all participating degrees of freedom are considered. However, any real physical system is not absolutely isolated, and will interact with its environment. This interaction with degrees of freedom external to

13230-441: The case of N {\displaystyle N} 3 {\displaystyle 3} -dimensional isotropic harmonic oscillators. That is, each particle in our ensemble can be treated as a simple harmonic oscillator . The Hamiltonian for this system is given by By using Hamilton's equations with the above Hamiltonian we find that the term in parentheses above is identically zero, thus yielding From this we can find

13377-543: The classical a priori probability . Liouville's theorem applies to conservative systems , that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system . Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general setting for conjugate position and momentum coordinates

13524-886: The concepts and notations of basic calculus , particularly derivatives with respect to space and time. A special case of the Schrödinger equation that admits a statement in those terms is the position-space Schrödinger equation for a single nonrelativistic particle in one dimension: i ℏ ∂ ∂ t Ψ ( x , t ) = [ − ℏ 2 2 m ∂ 2 ∂ x 2 + V ( x , t ) ] Ψ ( x , t ) . {\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)\right]\Psi (x,t).} Here, Ψ ( x , t ) {\displaystyle \Psi (x,t)}

13671-513: The definition of the chain rule. In this formalism, Liouville's Theorem states that the Lie derivative of the volume form is zero along the flow generated by X H {\displaystyle X_{H}} . That is, for ( M , ω ) {\displaystyle (M,\omega )} a 2n-dimensional symplectic manifold, In fact, the symplectic structure ω {\displaystyle \omega } itself

13818-623: The differential operator defined by p ^ x = − i ℏ d d x {\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}} the previous equation is evocative of the classic kinetic energy analogue , 1 2 m p ^ x 2 = E , {\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,} with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with

13965-483: The dynamics is trace-preserving and completely positive. The number of A m {\displaystyle A_{m}} operators is arbitrary, and they do not have to satisfy any special properties. But if the system is N {\displaystyle N} -dimensional, it can be shown that the master equation can be fully described by a set of N 2 − 1 {\displaystyle N^{2}-1} operators, provided they form

14112-598: The energy E {\displaystyle E} appears in the phase factor. This generalizes to any number of particles in any number of dimensions (in a time-independent potential): the standing wave solutions of the time-independent equation are the states with definite energy, instead of a probability distribution of different energies. In physics, these standing waves are called " stationary states " or " energy eigenstates "; in chemistry they are called " atomic orbitals " or " molecular orbitals ". Superpositions of energy eigenstates change their properties according to

14259-406: The equations of motion for the simple harmonic oscillator, these modified equations do not take the form of Hamilton's equations, and therefore we do not expect Liouville's theorem to hold. Instead, as depicted in the animation in this section, a generic phase space volume will shrink as it evolves under these equations of motion. To see this violation of Liouville's theorem explicitly, we can follow

14406-456: The evolution of d N {\displaystyle \mathrm {d} {\mathcal {N}}} particles. Within phase space, these d N {\displaystyle \mathrm {d} {\mathcal {N}}} particles occupy an infinitesimal volume given by We want d N d Γ {\displaystyle {\frac {\mathrm {d} {\mathcal {N}}}{\mathrm {d} \Gamma }}} to remain

14553-467: The evolution of ρ ( p , q ; t ) {\displaystyle \rho (p,q;t)} in time t {\displaystyle t} : Time derivatives are denoted by dots, and are evaluated according to Hamilton's equations for the system. This equation demonstrates the conservation of density in phase space (which was Gibbs 's name for the theorem). Liouville's theorem states that A proof of Liouville's theorem uses

14700-1603: The family U ^ ( t ) {\displaystyle {\hat {U}}(t)} . A Hamiltonian is just such a generator (up to the factor of the Planck constant that would be set to 1 in natural units ). To see that the generator is Hermitian, note that with U ^ ( δ t ) ≈ U ^ ( 0 ) − i G ^ δ t {\displaystyle {\hat {U}}(\delta t)\approx {\hat {U}}(0)-i{\hat {G}}\delta t} , we have U ^ ( δ t ) † U ^ ( δ t ) ≈ ( U ^ ( 0 ) † + i G ^ † δ t ) ( U ^ ( 0 ) − i G ^ δ t ) = I + i δ t ( G ^ † − G ^ ) + O ( δ t 2 ) , {\displaystyle {\hat {U}}(\delta t)^{\dagger }{\hat {U}}(\delta t)\approx ({\hat {U}}(0)^{\dagger }+i{\hat {G}}^{\dagger }\delta t)({\hat {U}}(0)-i{\hat {G}}\delta t)=I+i\delta t({\hat {G}}^{\dagger }-{\hat {G}})+O(\delta t^{2}),} so U ^ ( t ) {\displaystyle {\hat {U}}(t)}

14847-523: The first of these equations is consistent with the classical behavior, the second is not: If the pair ( ⟨ X ⟩ , ⟨ P ⟩ ) {\displaystyle (\langle X\rangle ,\langle P\rangle )} were to satisfy Newton's second law, the right-hand side of the second equation would have to be − V ′ ( ⟨ X ⟩ ) {\displaystyle -V'\left(\left\langle X\right\rangle \right)} which

14994-1294: The functions H n {\displaystyle {\mathcal {H}}_{n}} are the Hermite polynomials of order n {\displaystyle n} . The solution set may be generated by ψ n ( x ) = 1 n ! ( m ω 2 ℏ ) n ( x − ℏ m ω d d x ) n ( m ω π ℏ ) 1 4 e − m ω x 2 2 ℏ . {\displaystyle \psi _{n}(x)={\frac {1}{\sqrt {n!}}}\left({\sqrt {\frac {m\omega }{2\hbar }}}\right)^{n}\left(x-{\frac {\hbar }{m\omega }}{\frac {d}{dx}}\right)^{n}\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{4}}e^{\frac {-m\omega x^{2}}{2\hbar }}.} The eigenvalues are E n = ( n + 1 2 ) ℏ ω . {\displaystyle E_{n}=\left(n+{\frac {1}{2}}\right)\hbar \omega .} The case n = 0 {\displaystyle n=0}

15141-493: The general equation, or the specific nonrelativistic version. The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory , by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is an approximation that yields accurate results in many situations, but only to a certain extent (see relativistic quantum mechanics and relativistic quantum field theory ). To apply

15288-448: The infinitesimal volume of phase space: Thus we have ultimately found that the infinitesimal phase-space volume is unchanged, yielding demonstrating that Liouville's theorem holds for this system. The question remains of how the phase-space volume actually evolves in time. Above we have shown that the total volume is conserved, but said nothing about what it looks like. For a single particle we can see that its trajectory in phase space

15435-405: The interaction of a quantum system with its environment. One of these is the use of the density matrix , and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the Schrödinger picture or Heisenberg picture , it allows more easily for the inclusion of incoherent processes, which represent environmental interactions. The density operator has

15582-576: The interaction picture, defined by the unitary transformation M ~ = U 0 M U 0 † {\displaystyle {\tilde {M}}=U_{0}MU_{0}^{\dagger }} , where M {\displaystyle M} is an arbitrary operator, and U 0 = e i ( H S + H B ) t {\displaystyle U_{0}=e^{i(H_{S}+H_{B})t}} . Also note that U ( t , t 0 ) {\displaystyle U(t,t_{0})}

15729-692: The kinetic energy of the particle. The general solutions of the Schrödinger equation for the particle in a box are ψ ( x ) = A e i k x + B e − i k x E = ℏ 2 k 2 2 m {\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}} or, from Euler's formula , ψ ( x ) = C sin ⁡ ( k x ) + D cos ⁡ ( k x ) . {\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).} The infinite potential walls of

15876-505: The left side depends only on time; the one on the right side depends only on space. Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts Ψ ( r , t ) = ψ ( r ) τ ( t ) , {\displaystyle \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )\tau (t),} where ψ ( r ) {\displaystyle \psi (\mathbf {r} )}

16023-421: The linearity of ϕ t {\displaystyle \phi _{t}} , is a linear superoperator. The semigroup can be recovered as The Lindblad equation is invariant under any unitary transformation v of Lindblad operators and constants, and also under the inhomogeneous transformation where a i are complex numbers and b is a real number. However, the first transformation destroys

16170-448: The low density approximation, and the singular coupling limit. Each of these relies on specific physical assumptions regarding, e.g., correlation functions of the environment. For example, in the weak coupling limit derivation, one typically assumes that (a) correlations of the system with the environment develop slowly, (b) excitations of the environment caused by system decay quickly, and (c) terms which are fast-oscillating when compared to

16317-541: The momentum-space Schrödinger equation at each point in the Brillouin zone independently of the other points in the Brillouin zone. The Schrödinger equation is consistent with local probability conservation . It also ensures that a normalized wavefunction remains normalized after time evolution. In matrix mechanics, this means that the time evolution operator is a unitary operator . In contrast to, for example,

16464-525: The most common Lindblad equation describing the damping of a quantum harmonic oscillator (representing e.g. a Fabry–Perot cavity ) coupled to a thermal bath , with jump operators: Here n ¯ {\displaystyle {\overline {n}}} is the mean number of excitations in the reservoir damping the oscillator and γ is the decay rate. To model the quantum harmonic oscillator Hamiltonian with frequency ω c {\displaystyle \omega _{c}} of

16611-535: The motion through phase space as a 'fluid flow' of system points, the theorem that the convective derivative of the density, d ρ / d t {\displaystyle d\rho /dt} , is zero follows from the equation of continuity by noting that the 'velocity field' ( p ˙ , q ˙ ) {\displaystyle ({\dot {p}},{\dot {q}})} in phase space has zero divergence (which follows from Hamilton's relations). The theorem above

16758-529: The notes by Preskill , begins with a more general form of an open quantum system and converts it into Lindblad form by making the Markovian assumption and expanding in small time. A more physically motivated standard treatment covers three common types of derivations of the Lindbladian starting from a Hamiltonian acting on both the system and environment: the weak coupling limit (described in detail below),

16905-402: The orthonormality of the operators L i (unless all the γ i are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the γ i , the L i of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless. The Lindblad-type evolution of the density matrix in

17052-402: The parameter t {\displaystyle t} in such a way that U ^ ( t ) = e − i G ^ t {\displaystyle {\hat {U}}(t)=e^{-i{\hat {G}}t}} for some self-adjoint operator G ^ {\displaystyle {\hat {G}}} , called the generator of

17199-449: The particle exists. The constant i {\displaystyle i} is the imaginary unit , and ℏ {\displaystyle \hbar } is the reduced Planck constant , which has units of action ( energy multiplied by time). Broadening beyond this simple case, the mathematical formulation of quantum mechanics developed by Paul Dirac , David Hilbert , John von Neumann , and Hermann Weyl defines

17346-404: The phase-space density is not conserved. As can be seen from the equation as time increases, we expect our phase-space volume to decrease to zero as friction affects the system. As for how the phase-space volume evolves in time, we will still have the constant rotation as in the undamped case. However, the damping will introduce a steady decrease in the radii of each ellipse. Again we can solve for

17493-458: The photons, we can add a further unitary evolution: Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods. Schr%C3%B6dinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery

17640-618: The postulate that ψ {\displaystyle \psi } has norm 1. Therefore, since sin ⁡ ( k L ) = 0 {\displaystyle \sin(kL)=0} , k L {\displaystyle kL} must be an integer multiple of π {\displaystyle \pi } , k = n π L n = 1 , 2 , 3 , … . {\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .} This constraint on k {\displaystyle k} implies

17787-436: The potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common center of mass, and constitute a two-body problem to solve. The motion of the electron is of principal interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass. The Schrödinger equation for

17934-435: The probability ρ ( p , q ) d n q d n p {\displaystyle \rho (p,q)\;\mathrm {d} ^{n}q\,\mathrm {d} ^{n}p} that the system will be found in the infinitesimal phase space volume d n q d n p {\displaystyle \mathrm {d} ^{n}q\,\mathrm {d} ^{n}p} . The Liouville equation governs

18081-401: The property that it can represent a classical mixture of quantum states, and is thus vital to accurately describe the dynamics of so-called open quantum systems. The Lindblad master equation for system's density matrix ρ can be written as (for a pedagogical introduction you may refer to) where { a , b } = a b + b a {\displaystyle \{a,b\}=ab+ba}

18228-405: The relative phases between the energy levels. The energy eigenstates form a basis: any wave function may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is the spectral theorem in mathematics, and in a finite-dimensional state space it is just a statement of the completeness of the eigenvectors of

18375-614: The respective subspace of the total Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath. The most general form of this Hamiltonian is The dynamics of the entire system can be described by the Liouville equation of motion, χ ˙ = − i [ H , χ ] {\displaystyle {\dot {\chi }}=-i[H,\chi ]} . This equation, containing an infinite number of degrees of freedom,

18522-1002: The right hand side of the equation. If the interaction Hamiltonian is assumed to have the form for system operators α i {\displaystyle \alpha _{i}} and bath operators Γ i {\displaystyle \Gamma _{i}} then H ~ B S = ∑ i α ~ i Γ ~ i {\displaystyle {\tilde {H}}_{BS}=\sum _{i}{\tilde {\alpha }}_{i}{\tilde {\Gamma }}_{i}} . The master equation becomes which can be expanded as The expectation values ⟨ Γ i Γ j ⟩ = tr ⁡ { Γ i Γ j R 0 } {\displaystyle \langle \Gamma _{i}\Gamma _{j}\rangle =\operatorname {tr} \{\Gamma _{i}\Gamma _{j}R_{0}\}} are with respect to

18669-734: The role of velocity, it does not represent velocity at a point since simultaneous measurement of position and velocity violates uncertainty principle . If the Hamiltonian is not an explicit function of time, Schrödinger's equation reads: i ℏ ∂ ∂ t Ψ ( r , t ) = [ − ℏ 2 2 m ∇ 2 + V ( r ) ] Ψ ( r , t ) . {\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right]\Psi (\mathbf {r} ,t).} The operator on

18816-1058: The same throughout time, so that ρ ( Γ , t ) {\displaystyle \rho (\Gamma ,t)} is constant along the trajectories of the system. If we allow our particles to evolve by an infinitesimal time step δ t {\displaystyle \delta t} , we see that each particle phase space location changes as where q i ˙ {\displaystyle {\dot {q_{i}}}} and p i ˙ {\displaystyle {\dot {p_{i}}}} denote d q i d t {\displaystyle {\frac {dq_{i}}{dt}}} and d p i d t {\displaystyle {\frac {dp_{i}}{dt}}} respectively, and we have only kept terms linear in δ t {\displaystyle \delta t} . Extending this to our infinitesimal hypercube d Γ {\displaystyle \mathrm {d} \Gamma } ,

18963-729: The same, since both will be approximately equal to V ′ ( x 0 ) {\displaystyle V'(x_{0})} . In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position. The Schrödinger equation in its general form i ℏ ∂ ∂ t Ψ ( r , t ) = H ^ Ψ ( r , t ) {\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi \left(\mathbf {r} ,t\right)={\hat {H}}\Psi \left(\mathbf {r} ,t\right)}

19110-541: The second derivative becomes the Laplacian ∇ 2 {\displaystyle \nabla ^{2}} . The canonical commutation relation also implies that the position and momentum operators are Fourier conjugates of each other. Consequently, functions originally defined in terms of their position dependence can be converted to functions of momentum using the Fourier transform. In solid-state physics ,

19257-420: The side lengths change as To find the new infinitesimal phase-space volume d Γ ′ {\displaystyle \mathrm {d} \Gamma '} , we need the product of the above quantities. To first order in δ t {\displaystyle \delta t} , we get the following: So far, we have yet to make any specifications about our system. Let us now specialize to

19404-567: The state of a quantum mechanical system to be a vector | ψ ⟩ {\displaystyle |\psi \rangle } belonging to a separable complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector is postulated to be normalized under the Hilbert space's inner product, that is, in Dirac notation it obeys ⟨ ψ | ψ ⟩ = 1 {\displaystyle \langle \psi |\psi \rangle =1} . The exact nature of this Hilbert space

19551-529: The superoperator L {\displaystyle {\mathcal {L}}} is called the Lindbladian superoperator or the Liouvillian superoperator . More generally, the GKSL equation has the form where { A m } {\displaystyle \{A_{m}\}} are arbitrary operators and h is a positive semidefinite matrix. The latter is a strict requirement to ensure

19698-667: The system density matrix but requires full knowledge of the dynamics of the bath degrees of freedom. A simplifying assumption called the Born approximation rests on the largeness of the bath and the relative weakness of the coupling, which is to say the coupling of the system to the bath should not significantly alter the bath eigenstates. In this case the full density matrix is factorable for all times as χ ~ ( t ) = ρ ~ ( t ) R 0 {\displaystyle {\tilde {\chi }}(t)={\tilde {\rho }}(t)R_{0}} . The master equation becomes The equation

19845-449: The system results in dissipation of energy into the surroundings, causing decay and randomization of phase. More so, understanding the interaction of a quantum system with its environment is necessary for understanding many commonly observed phenomena like the spontaneous emission of light from excited atoms, or the performance of many quantum technological devices, like the laser. Certain mathematical techniques have been introduced to treat

19992-403: The system timescale of interest can be neglected. These three approximations are called Born, Markov, and rotating wave, respectively. The weak-coupling limit derivation assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a Hamiltonian written in terms of operators acting only on

20139-417: The system under investigation is a part of a larger whole, density matrices may be used instead. A density matrix is a positive semi-definite operator whose trace is equal to 1. (The term "density operator" is also used, particularly when the underlying Hilbert space is infinite-dimensional.) The set of all density matrices is convex , and the extreme points are the operators that project onto vectors in

20286-607: The system-environment dynamics, or phenomenologically modelled . γ i ≥ 0 {\displaystyle \gamma _{i}\geq 0} are a set of non-negative real coefficients called damping rates . If all γ i = 0 {\displaystyle \gamma _{i}=0} one recovers the von Neumann equation ρ ˙ = − ( i / ℏ ) [ H , ρ ] {\displaystyle {\dot {\rho }}=-(i/\hbar )[H,\rho ]} describing unitary dynamics, which

20433-472: The task of solving the time-dependent Schrödinger equation for any state. Stationary states can also be described by a simpler form of the Schrödinger equation, the time-independent Schrödinger equation. H ^ ⁡ | Ψ ⟩ = E | Ψ ⟩ {\displaystyle \operatorname {\hat {H}} |\Psi \rangle =E|\Psi \rangle } where E {\displaystyle E}

20580-437: The time evolution of the phase space distribution function . Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics. It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838. Consider

20727-414: The trajectories explicitly using Hamilton's equations, taking care to use the modified ones above. Letting α ≡ γ 2 {\displaystyle \alpha \equiv {\frac {\gamma }{2}}} for convenience, we find where the values Q i {\displaystyle Q_{i}} and P i {\displaystyle P_{i}} denote

20874-402: The two state vectors where a and b are any complex numbers. Moreover, the sum can be extended for any number of state vectors. This property allows superpositions of quantum states to be solutions of the Schrödinger equation. Even more generally, it holds that a general solution to the Schrödinger equation can be found by taking a weighted sum over a basis of states. A choice often employed

21021-692: The unitary operator U ^ ( t ) {\displaystyle {\hat {U}}(t)} describes wave function evolution over some time interval, then the time evolution of a density matrix over that same interval is given by ρ ^ ( t ) = U ^ ( t ) ρ ^ ( 0 ) U ^ ( t ) † . {\displaystyle {\hat {\rho }}(t)={\hat {U}}(t){\hat {\rho }}(0){\hat {U}}(t)^{\dagger }.} Liouville%27s theorem (Hamiltonian) In physics , Liouville's theorem , named after

21168-467: The vectors | ψ E n ⟩ {\displaystyle |\psi _{E_{n}}\rangle } are solutions of the time-independent equation H ^ | ψ E n ⟩ = E n | ψ E n ⟩ {\displaystyle {\hat {H}}|\psi _{E_{n}}\rangle =E_{n}|\psi _{E_{n}}\rangle } . Holding

21315-741: The well. Another related problem is that of the rectangular potential barrier , which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy . The Schrödinger equation for this situation is E ψ = − ℏ 2 2 m d 2 d x 2 ψ + 1 2 m ω 2 x 2 ψ , {\displaystyle E\psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi +{\frac {1}{2}}m\omega ^{2}x^{2}\psi ,} where x {\displaystyle x}

21462-499: Was a significant landmark in the development of quantum mechanics . It is named after Erwin Schrödinger , who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics . Given a set of known initial conditions, Newton's second law makes

21609-461: Was fixed by Dirac by taking the so-called square-root of the Klein-Gordon operator and in turn introducing Dirac matrices . In a modern context, the Klein-Gordon equation describes spin-less particles, while the Dirac equation describes spin-1/2 particles. Introductory courses on physics or chemistry typically introduce the Schrödinger equation in a way that can be appreciated knowing only

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