Misplaced Pages

#625374

73-415: The LLCD pulsed laser system conducted a successful test on October 18, 2013, transmitting data between the spacecraft and its ground station on Earth at a distance of 385,000 kilometres (239,000 mi). This test set a downlink record of 622 megabits per second (Mbps) from spacecraft to ground, and an "error-free data upload rate of 20 Mbps" from ground station to spacecraft. Tests were carried out over

146-455: A + a † ) {\displaystyle {\hat {x}}={\sqrt {\frac {\hbar }{2m\omega }}}(a+a^{\dagger })} p ^ = i m ω ℏ 2 ( a † − a ) . {\displaystyle {\hat {p}}=i{\sqrt {\frac {m\omega \hbar }{2}}}(a^{\dagger }-a).} Using the standard rules for creation and annihilation operators on

219-509: A gain medium must have a gain bandwidth sufficiently broad to amplify those frequencies. An example of a suitable material is titanium -doped, artificially grown sapphire ( Ti:sapphire ) which has a very wide gain bandwidth and can thus produce pulses of only a few femtoseconds duration. Such mode-locked lasers are a most versatile tool for researching processes occurring on extremely short time scales (known as femtosecond physics, femtosecond chemistry and ultrafast science ), for maximizing

292-461: A vector in a function space . We can define an inner product for a pair of functions u ( x ) and v ( x ) in this vector space: ⟨ u ∣ v ⟩ = ∫ − ∞ ∞ u ∗ ( x ) ⋅ v ( x ) d x , {\displaystyle \langle u\mid v\rangle =\int _{-\infty }^{\infty }u^{*}(x)\cdot v(x)\,dx,} where

365-1711: A "balanced" way. Moreover, every squeezed coherent state also saturates the Kennard bound although the individual contributions of position and momentum need not be balanced in general. Consider a particle in a one-dimensional box of length L {\displaystyle L} . The eigenfunctions in position and momentum space are ψ n ( x , t ) = { A sin ⁡ ( k n x ) e − i ω n t , 0 < x < L , 0 , otherwise, {\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin(k_{n}x)\mathrm {e} ^{-\mathrm {i} \omega _{n}t},&0<x<L,\\0,&{\text{otherwise,}}\end{cases}}} and φ n ( p , t ) = π L ℏ n ( 1 − ( − 1 ) n e − i k L ) e − i ω n t π 2 n 2 − k 2 L 2 , {\displaystyle \varphi _{n}(p,t)={\sqrt {\frac {\pi L}{\hbar }}}\,\,{\frac {n\left(1-(-1)^{n}e^{-ikL}\right)e^{-i\omega _{n}t}}{\pi ^{2}n^{2}-k^{2}L^{2}}},} where ω n = π 2 ℏ n 2 8 L 2 m {\textstyle \omega _{n}={\frac {\pi ^{2}\hbar n^{2}}{8L^{2}m}}} and we have used

438-520: A 30-day test period. The LLCD is a free-space optical communication system. It was NASA's first attempt at two-way space communication using an optical laser instead of radio waves . It is expected to lead to operational laser systems on future NASA satellites. The next iteration of the concept will be the Laser Communications Relay Demonstration scheduled for 2017. Also, it has been proposed as payload for

511-439: A cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta. One way to quantify the precision of the position and momentum is the standard deviation   σ . Since | ψ ( x ) | 2 {\displaystyle |\psi (x)|^{2}} is a probability density function for position, we calculate its standard deviation. The precision of

584-569: A formulation for arbitrary Hermitian operator operators O ^ {\displaystyle {\hat {\mathcal {O}}}} expressed in terms of their standard deviation σ O = ⟨ O ^ 2 ⟩ − ⟨ O ^ ⟩ 2 , {\displaystyle \sigma _{\mathcal {O}}={\sqrt {\langle {\hat {\mathcal {O}}}^{2}\rangle -\langle {\hat {\mathcal {O}}}\rangle ^{2}}},} where

657-506: A given pulse energy, this requires creating pulses of the shortest possible duration utilizing techniques such as Q-switching . The optical bandwidth of a pulse cannot be narrower than the reciprocal of the pulse width. In the case of extremely short pulses, that implies lasing over a considerable bandwidth, quite contrary to the very narrow bandwidths typical of continuous wave (CW) lasers. The lasing medium in some dye lasers and vibronic solid-state lasers produces optical gain over

730-444: A laser beam comes into contact with soft-tissue, one important factor is to not overheat surrounding tissue, so necrosis can be prevented. Laser pulses must be spaced out to allow for efficient tissue cooling (thermal relaxation time) between pulses. Uncertainty principle The uncertainty principle , also known as Heisenberg's indeterminacy principle , is a fundamental concept in quantum mechanics . It states that there

803-460: A number of different motivations. Some lasers are pulsed simply because they cannot be run in continuous mode. In other cases the application requires the production of pulses having as large an energy as possible. Since the pulse energy is equal to the average power divided by the repetition rate, this goal can sometimes be satisfied by lowering the rate of pulses so that more energy can be built up in between pulses. In laser ablation for example,

SECTION 10

#1732779920626

876-664: A particle initially has a momentum space wave function described by a normal distribution around some constant momentum p 0 according to φ ( p ) = ( x 0 ℏ π ) 1 / 2 exp ⁡ ( − x 0 2 ( p − p 0 ) 2 2 ℏ 2 ) , {\displaystyle \varphi (p)=\left({\frac {x_{0}}{\hbar {\sqrt {\pi }}}}\right)^{1/2}\exp \left({\frac {-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}\right),} where we have introduced

949-858: A particle moving along with constant momentum at arbitrarily high precision. On the other hand, the standard deviation of the position is σ x = x 0 2 1 + ω 0 2 t 2 {\displaystyle \sigma _{x}={\frac {x_{0}}{\sqrt {2}}}{\sqrt {1+\omega _{0}^{2}t^{2}}}} such that the uncertainty product can only increase with time as σ x ( t ) σ p ( t ) = ℏ 2 1 + ω 0 2 t 2 {\displaystyle \sigma _{x}(t)\sigma _{p}(t)={\frac {\hbar }{2}}{\sqrt {1+\omega _{0}^{2}t^{2}}}} Starting with Kennard's derivation of position-momentum uncertainty, Howard Percy Robertson developed

1022-406: A position and a momentum eigenstate. When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is not a momentum eigenstate, however, but rather it can be represented as a sum of multiple momentum basis eigenstates. In other words,

1095-741: A quantum system, such as position , x , and momentum, p . Such paired-variables are known as complementary variables or canonically conjugate variables . First introduced in 1927 by German physicist Werner Heisenberg , the formal inequality relating the standard deviation of position σ x and the standard deviation of momentum σ p was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928: σ x σ p ≥ ℏ 2 {\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}} where ℏ = h 2 π {\displaystyle \hbar ={\frac {h}{2\pi }}}

1168-2299: A reference scale x 0 = ℏ / m ω 0 {\textstyle x_{0}={\sqrt {\hbar /m\omega _{0}}}} , with ω 0 > 0 {\displaystyle \omega _{0}>0} describing the width of the distribution—cf. nondimensionalization . If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are Φ ( p , t ) = ( x 0 ℏ π ) 1 / 2 exp ⁡ ( − x 0 2 ( p − p 0 ) 2 2 ℏ 2 − i p 2 t 2 m ℏ ) , {\displaystyle \Phi (p,t)=\left({\frac {x_{0}}{\hbar {\sqrt {\pi }}}}\right)^{1/2}\exp \left({\frac {-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}-{\frac {ip^{2}t}{2m\hbar }}\right),} Ψ ( x , t ) = ( 1 x 0 π ) 1 / 2 e − x 0 2 p 0 2 / 2 ℏ 2 1 + i ω 0 t exp ⁡ ( − ( x − i x 0 2 p 0 / ℏ ) 2 2 x 0 2 ( 1 + i ω 0 t ) ) . {\displaystyle \Psi (x,t)=\left({\frac {1}{x_{0}{\sqrt {\pi }}}}\right)^{1/2}{\frac {e^{-x_{0}^{2}p_{0}^{2}/2\hbar ^{2}}}{\sqrt {1+i\omega _{0}t}}}\,\exp \left(-{\frac {(x-ix_{0}^{2}p_{0}/\hbar )^{2}}{2x_{0}^{2}(1+i\omega _{0}t)}}\right).} Since ⟨ p ( t ) ⟩ = p 0 {\displaystyle \langle p(t)\rangle =p_{0}} and σ p ( t ) = ℏ / ( 2 x 0 ) {\displaystyle \sigma _{p}(t)=\hbar /({\sqrt {2}}x_{0})} , this can be interpreted as

1241-1186: A right eigenstate of position with a constant eigenvalue x 0 . By definition, this means that x ^ | ψ ⟩ = x 0 | ψ ⟩ . {\displaystyle {\hat {x}}|\psi \rangle =x_{0}|\psi \rangle .} Applying the commutator to | ψ ⟩ {\displaystyle |\psi \rangle } yields [ x ^ , p ^ ] | ψ ⟩ = ( x ^ p ^ − p ^ x ^ ) | ψ ⟩ = ( x ^ − x 0 I ^ ) p ^ | ψ ⟩ = i ℏ | ψ ⟩ , {\displaystyle [{\hat {x}},{\hat {p}}]|\psi \rangle =({\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}})|\psi \rangle =({\hat {x}}-x_{0}{\hat {I}}){\hat {p}}\,|\psi \rangle =i\hbar |\psi \rangle ,} where Î

1314-443: A small volume of material at the surface of a work piece can be evaporated if it is heated in a very short time, whereas supplying the energy gradually would allow for the heat to be absorbed into the bulk of the piece, never attaining a sufficiently high temperature at a particular point. Other applications rely on the peak pulse power (rather than the energy in the pulse), especially in order to obtain nonlinear optical effects. For

1387-476: A way that generalizes more easily. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables ). A nonzero function and its Fourier transform cannot both be sharply localized at

1460-423: A wide bandwidth, making a laser possible which can thus generate pulses of light as short as a few femtoseconds . In a Q-switched laser, the population inversion is allowed to build up by introducing loss inside the resonator which exceeds the gain of the medium; this can also be described as a reduction of the quality factor or 'Q' of the cavity. Then, after the pump energy stored in the laser medium has approached

1533-403: Is ψ ( x ) ∝ e i k 0 x = e i p 0 x / ℏ   . {\displaystyle \psi (x)\propto e^{ik_{0}x}=e^{ip_{0}x/\hbar }~.} The Born rule states that this should be interpreted as a probability density amplitude function in the sense that the probability of finding

SECTION 20

#1732779920626

1606-600: Is 1 if X = x {\displaystyle X=x} and 0 otherwise. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. On the other hand, consider a wave function that is a sum of many waves , which we may write as ψ ( x ) ∝ ∑ n A n e i p n x / ℏ   , {\displaystyle \psi (x)\propto \sum _{n}A_{n}e^{ip_{n}x/\hbar }~,} where A n represents

1679-452: Is a limit to the precision with which certain pairs of physical properties, such as position and momentum , can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known. More formally, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on

1752-731: Is a right eigenstate of the annihilation operator , a ^ | α ⟩ = α | α ⟩ , {\displaystyle {\hat {a}}|\alpha \rangle =\alpha |\alpha \rangle ,} which may be represented in terms of Fock states as | α ⟩ = e − | α | 2 2 ∑ n = 0 ∞ α n n ! | n ⟩ {\displaystyle |\alpha \rangle =e^{-{|\alpha |^{2} \over 2}}\sum _{n=0}^{\infty }{\alpha ^{n} \over {\sqrt {n!}}}|n\rangle } In

1825-511: Is also required for three-level lasers in which the lower energy level rapidly becomes highly populated preventing further lasing until those atoms relax to the ground state. These lasers, such as the excimer laser and the copper vapor laser, can never be operated in CW mode. Pulsed Nd:YAG and Er:YAG lasers are used in laser tattoo removal and laser range finders among other applications. Pulsed lasers are also used in soft-tissue surgery. When

1898-687: Is just the normal distribution . In a quantum harmonic oscillator of characteristic angular frequency ω , place a state that is offset from the bottom of the potential by some displacement x 0 as ψ ( x ) = ( m Ω π ℏ ) 1 / 4 exp ⁡ ( − m Ω ( x − x 0 ) 2 2 ℏ ) , {\displaystyle \psi (x)=\left({\frac {m\Omega }{\pi \hbar }}\right)^{1/4}\exp {\left(-{\frac {m\Omega (x-x_{0})^{2}}{2\hbar }}\right)},} where Ω describes

1971-414: Is not in an eigenstate of that observable. The uncertainty principle can be visualized using the position- and momentum-space wavefunctions for one spinless particle with mass in one dimension. The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentum-space wavefunction is less localized so

2044-493: Is the commutator . For a pair of operators  and B ^ {\displaystyle {\hat {B}}} , one defines their commutator as [ A ^ , B ^ ] = A ^ B ^ − B ^ A ^ . {\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}.} In

2117-1014: Is the identity operator . Suppose, for the sake of proof by contradiction , that | ψ ⟩ {\displaystyle |\psi \rangle } is also a right eigenstate of momentum, with constant eigenvalue p 0 . If this were true, then one could write ( x ^ − x 0 I ^ ) p ^ | ψ ⟩ = ( x ^ − x 0 I ^ ) p 0 | ψ ⟩ = ( x 0 I ^ − x 0 I ^ ) p 0 | ψ ⟩ = 0. {\displaystyle ({\hat {x}}-x_{0}{\hat {I}}){\hat {p}}\,|\psi \rangle =({\hat {x}}-x_{0}{\hat {I}})p_{0}\,|\psi \rangle =(x_{0}{\hat {I}}-x_{0}{\hat {I}})p_{0}\,|\psi \rangle =0.} On

2190-476: Is the reduced Planck constant . The quintessentially quantum mechanical uncertainty principle comes in many forms other than position–momentum. The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time. The basic principle has been extended in numerous directions; it must be considered in many kinds of fundamental physical measurements. It

2263-421: Is then σ x σ p = ℏ ( n + 1 2 ) ≥ ℏ 2 .   {\displaystyle \sigma _{x}\sigma _{p}=\hbar \left(n+{\frac {1}{2}}\right)\geq {\frac {\hbar }{2}}.~} In particular, the above Kennard bound is saturated for the ground state n =0 , for which the probability density

Lunar Laser Communication Demonstration - Misplaced Pages Continue

2336-2742: Is to evaluate these inner products. ⟨ f ∣ g ⟩ − ⟨ g ∣ f ⟩ = ∫ − ∞ ∞ ψ ∗ ( x ) x ⋅ ( − i ℏ d d x ) ψ ( x ) d x − ∫ − ∞ ∞ ψ ∗ ( x ) ( − i ℏ d d x ) ⋅ x ψ ( x ) d x = i ℏ ⋅ ∫ − ∞ ∞ ψ ∗ ( x ) [ ( − x ⋅ d ψ ( x ) d x ) + d ( x ψ ( x ) ) d x ] d x = i ℏ ⋅ ∫ − ∞ ∞ ψ ∗ ( x ) [ ( − x ⋅ d ψ ( x ) d x ) + ψ ( x ) + ( x ⋅ d ψ ( x ) d x ) ] d x = i ℏ ⋅ ∫ − ∞ ∞ ψ ∗ ( x ) ψ ( x ) d x = i ℏ ⋅ ∫ − ∞ ∞ | ψ ( x ) | 2 d x = i ℏ {\displaystyle {\begin{aligned}\langle f\mid g\rangle -\langle g\mid f\rangle &=\int _{-\infty }^{\infty }\psi ^{*}(x)\,x\cdot \left(-i\hbar {\frac {d}{dx}}\right)\,\psi (x)\,dx-\int _{-\infty }^{\infty }\psi ^{*}(x)\,\left(-i\hbar {\frac {d}{dx}}\right)\cdot x\,\psi (x)\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }\psi ^{*}(x)\left[\left(-x\cdot {\frac {d\psi (x)}{dx}}\right)+{\frac {d(x\psi (x))}{dx}}\right]\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }\psi ^{*}(x)\left[\left(-x\cdot {\frac {d\psi (x)}{dx}}\right)+\psi (x)+\left(x\cdot {\frac {d\psi (x)}{dx}}\right)\right]\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }\psi ^{*}(x)\psi (x)\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }|\psi (x)|^{2}\,dx\\&=i\hbar \end{aligned}}} Plugging this into

2409-507: Is to pump the laser material with a source that is itself pulsed, either through electronic charging in the case of flash lamps, or another laser which is already pulsed. Pulsed pumping was historically used with dye lasers where the inverted population lifetime of a dye molecule was so short that a high energy, fast pump was needed. The way to overcome this problem was to charge up large capacitors which are then switched to discharge through flashlamps , producing an intense flash. Pulsed pumping

2482-431: Is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic scales that humans experience. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, but the more abstract matrix mechanics picture formulates it in

2555-494: The Phobos And Deimos & Mars Environment (PADME) orbiter. This space - or spaceflight -related article is a stub . You can help Misplaced Pages by expanding it . Pulsed laser Pulsed operation of lasers refers to any laser not classified as continuous wave , so that the optical power appears in pulses of some duration at some repetition rate . This encompasses a wide range of technologies addressing

2628-764: The de Broglie relation p = ℏ k {\displaystyle p=\hbar k} . The variances of x {\displaystyle x} and p {\displaystyle p} can be calculated explicitly: σ x 2 = L 2 12 ( 1 − 6 n 2 π 2 ) {\displaystyle \sigma _{x}^{2}={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)} σ p 2 = ( ℏ n π L ) 2 . {\displaystyle \sigma _{p}^{2}=\left({\frac {\hbar n\pi }{L}}\right)^{2}.} The product of

2701-1008: The means vanish, which just amounts to a shift of the origin of our coordinates. (A more general proof that does not make this assumption is given below.) This gives us the simpler form σ x 2 = ∫ − ∞ ∞ x 2 ⋅ | ψ ( x ) | 2 d x {\displaystyle \sigma _{x}^{2}=\int _{-\infty }^{\infty }x^{2}\cdot |\psi (x)|^{2}\,dx} σ p 2 = ∫ − ∞ ∞ p 2 ⋅ | φ ( p ) | 2 d p   . {\displaystyle \sigma _{p}^{2}=\int _{-\infty }^{\infty }p^{2}\cdot |\varphi (p)|^{2}\,dp~.} The function f ( x ) = x ⋅ ψ ( x ) {\displaystyle f(x)=x\cdot \psi (x)} can be interpreted as

2774-519: The Kennard bound σ x σ p = ℏ 2 m ω ℏ m ω 2 = ℏ 2 . {\displaystyle \sigma _{x}\sigma _{p}={\sqrt {\frac {\hbar }{2m\omega }}}\,{\sqrt {\frac {\hbar m\omega }{2}}}={\frac {\hbar }{2}}.} with position and momentum each contributing an amount ℏ / 2 {\textstyle {\sqrt {\hbar /2}}} in

2847-790: The Robertson uncertainty relation is given by σ A σ B ≥ | 1 2 i ⟨ [ A ^ , B ^ ] ⟩ | = 1 2 | ⟨ [ A ^ , B ^ ] ⟩ | . {\displaystyle \sigma _{A}\sigma _{B}\geq \left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|={\frac {1}{2}}\left|\langle [{\hat {A}},{\hat {B}}]\rangle \right|.} Erwin Schrödinger showed how to allow for correlation between

2920-749: The above inequalities, we get σ x 2 σ p 2 ≥ | ⟨ f ∣ g ⟩ | 2 ≥ ( ⟨ f ∣ g ⟩ − ⟨ g ∣ f ⟩ 2 i ) 2 = ( i ℏ 2 i ) 2 = ℏ 2 4 {\displaystyle \sigma _{x}^{2}\sigma _{p}^{2}\geq |\langle f\mid g\rangle |^{2}\geq \left({\frac {\langle f\mid g\rangle -\langle g\mid f\rangle }{2i}}\right)^{2}=\left({\frac {i\hbar }{2i}}\right)^{2}={\frac {\hbar ^{2}}{4}}} or taking

2993-454: The amplitude of these modes and is called the wave function in momentum space . In mathematical terms, we say that φ ( p ) {\displaystyle \varphi (p)} is the Fourier transform of ψ ( x ) {\displaystyle \psi (x)} and that x and p are conjugate variables . Adding together all of these plane waves comes at

Lunar Laser Communication Demonstration - Misplaced Pages Continue

3066-544: The asterisk denotes the complex conjugate . With this inner product defined, we note that the variance for position can be written as σ x 2 = ∫ − ∞ ∞ | f ( x ) | 2 d x = ⟨ f ∣ f ⟩   . {\displaystyle \sigma _{x}^{2}=\int _{-\infty }^{\infty }|f(x)|^{2}\,dx=\langle f\mid f\rangle ~.} We can repeat this for momentum by interpreting

3139-864: The brackets ⟨ O ^ ⟩ {\displaystyle \langle {\hat {\mathcal {O}}}\rangle } indicate an expectation value of the observable represented by operator O ^ {\displaystyle {\hat {\mathcal {O}}}} . For a pair of operators A ^ {\displaystyle {\hat {A}}} and B ^ {\displaystyle {\hat {B}}} , define their commutator as [ A ^ , B ^ ] = A ^ B ^ − B ^ A ^ , {\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}},} and

3212-509: The case of position and momentum, the commutator is the canonical commutation relation [ x ^ , p ^ ] = i ℏ . {\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar .} The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum eigenstates . Let | ψ ⟩ {\displaystyle |\psi \rangle } be

3285-409: The effect of nonlinearity in optical materials (e.g. in second-harmonic generation , parametric down-conversion , optical parametric oscillators and the like) due to the large peak power, and in ablation applications. Again, because of the extremely short pulse duration, such a laser will produce pulses which achieve an extremely high peak power. Another method of achieving pulsed laser operation

3358-395: The energy eigenstates, a † | n ⟩ = n + 1 | n + 1 ⟩ {\displaystyle a^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle } a | n ⟩ = n | n − 1 ⟩ , {\displaystyle a|n\rangle ={\sqrt {n}}|n-1\rangle ,}

3431-469: The equation above to get | ⟨ f ∣ g ⟩ | 2 ≥ ( ⟨ f ∣ g ⟩ − ⟨ g ∣ f ⟩ 2 i ) 2   . {\displaystyle |\langle f\mid g\rangle |^{2}\geq \left({\frac {\langle f\mid g\rangle -\langle g\mid f\rangle }{2i}}\right)^{2}~.} All that remains

3504-927: The following (the right most equality holds only when Ω = ω ): σ x σ p ≥ ℏ 4 3 + 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) − ( 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) − 1 ) = ℏ 2 . {\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-1\right)}}={\frac {\hbar }{2}}.} A coherent state

3577-452: The function g ~ ( p ) = p ⋅ φ ( p ) {\displaystyle {\tilde {g}}(p)=p\cdot \varphi (p)} as a vector, but we can also take advantage of the fact that ψ ( x ) {\displaystyle \psi (x)} and φ ( p ) {\displaystyle \varphi (p)} are Fourier transforms of each other. We evaluate

3650-482: The integration by parts, the cancelled term vanishes because the wave function vanishes at infinity, and the final two integrations re-assert the Fourier transforms. Often the term − i ℏ d d x {\textstyle -i\hbar {\frac {d}{dx}}} is called the momentum operator in position space. Applying Plancherel's theorem and then Parseval's theorem , we see that

3723-4366: The inverse Fourier transform through integration by parts : g ( x ) = 1 2 π ℏ ⋅ ∫ − ∞ ∞ g ~ ( p ) ⋅ e i p x / ℏ d p = 1 2 π ℏ ∫ − ∞ ∞ p ⋅ φ ( p ) ⋅ e i p x / ℏ d p = 1 2 π ℏ ∫ − ∞ ∞ [ p ⋅ ∫ − ∞ ∞ ψ ( χ ) e − i p χ / ℏ d χ ] ⋅ e i p x / ℏ d p = i 2 π ∫ − ∞ ∞ [ ψ ( χ ) e − i p χ / ℏ | − ∞ ∞ − ∫ − ∞ ∞ d ψ ( χ ) d χ e − i p χ / ℏ d χ ] ⋅ e i p x / ℏ d p = − i ∫ − ∞ ∞ d ψ ( χ ) d χ [ 1 2 π ∫ − ∞ ∞ e i p ( x − χ ) / ℏ d p ] d χ = − i ∫ − ∞ ∞ d ψ ( χ ) d χ [ δ ( x − χ ℏ ) ] d χ = − i ℏ ∫ − ∞ ∞ d ψ ( χ ) d χ [ δ ( x − χ ) ] d χ = − i ℏ d ψ ( x ) d x = ( − i ℏ d d x ) ⋅ ψ ( x ) , {\displaystyle {\begin{aligned}g(x)&={\frac {1}{\sqrt {2\pi \hbar }}}\cdot \int _{-\infty }^{\infty }{\tilde {g}}(p)\cdot e^{ipx/\hbar }\,dp\\&={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }p\cdot \varphi (p)\cdot e^{ipx/\hbar }\,dp\\&={\frac {1}{2\pi \hbar }}\int _{-\infty }^{\infty }\left[p\cdot \int _{-\infty }^{\infty }\psi (\chi )e^{-ip\chi /\hbar }\,d\chi \right]\cdot e^{ipx/\hbar }\,dp\\&={\frac {i}{2\pi }}\int _{-\infty }^{\infty }\left[{\cancel {\left.\psi (\chi )e^{-ip\chi /\hbar }\right|_{-\infty }^{\infty }}}-\int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}e^{-ip\chi /\hbar }\,d\chi \right]\cdot e^{ipx/\hbar }\,dp\\&=-i\int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\,e^{ip(x-\chi )/\hbar }\,dp\right]\,d\chi \\&=-i\int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}\left[\delta \left({\frac {x-\chi }{\hbar }}\right)\right]\,d\chi \\&=-i\hbar \int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}\left[\delta \left(x-\chi \right)\right]\,d\chi \\&=-i\hbar {\frac {d\psi (x)}{dx}}\\&=\left(-i\hbar {\frac {d}{dx}}\right)\cdot \psi (x),\end{aligned}}} where v = ℏ − i p e − i p χ / ℏ {\displaystyle v={\frac {\hbar }{-ip}}e^{-ip\chi /\hbar }} in

SECTION 50

#1732779920626

3796-412: The maximum possible level, the introduced loss mechanism (often an electro- or acousto-optical element) is rapidly removed (or that occurs by itself in a passive device), allowing lasing to begin which rapidly obtains the stored energy in the gain medium. This results in a short pulse incorporating that energy, and thus a high peak power. A mode-locked laser is capable of emitting extremely short pulses on

3869-733: The momentum must be less precise. This precision may be quantified by the standard deviations, σ x = ⟨ x ^ 2 ⟩ − ⟨ x ^ ⟩ 2 {\displaystyle \sigma _{x}={\sqrt {\langle {\hat {x}}^{2}\rangle -\langle {\hat {x}}\rangle ^{2}}}} σ p = ⟨ p ^ 2 ⟩ − ⟨ p ^ ⟩ 2 . {\displaystyle \sigma _{p}={\sqrt {\langle {\hat {p}}^{2}\rangle -\langle {\hat {p}}\rangle ^{2}}}.} As in

3942-1948: The notation N ( μ , σ 2 ) {\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} to denote a normal distribution of mean μ and variance σ . Copying the variances above and applying trigonometric identities , we can write the product of the standard deviations as σ x σ p = ℏ 2 ( cos 2 ⁡ ( ω t ) + Ω 2 ω 2 sin 2 ⁡ ( ω t ) ) ( cos 2 ⁡ ( ω t ) + ω 2 Ω 2 sin 2 ⁡ ( ω t ) ) = ℏ 4 3 + 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) − ( 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) − 1 ) cos ⁡ ( 4 ω t ) {\displaystyle {\begin{aligned}\sigma _{x}\sigma _{p}&={\frac {\hbar }{2}}{\sqrt {\left(\cos ^{2}{(\omega t)}+{\frac {\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega t)}\right)\left(\cos ^{2}{(\omega t)}+{\frac {\omega ^{2}}{\Omega ^{2}}}\sin ^{2}{(\omega t)}\right)}}\\&={\frac {\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-1\right)\cos {(4\omega t)}}}\end{aligned}}} From

4015-456: The order of tens of picoseconds down to less than 10 femtoseconds . These pulses will repeat at the round trip time, that is, the time that it takes light to complete one round trip between the mirrors comprising the resonator. Due to the Fourier limit (also known as energy-time uncertainty ), a pulse of such short temporal length has a spectrum spread over a considerable bandwidth. Thus such

4088-424: The other hand, the above canonical commutation relation requires that [ x ^ , p ^ ] | ψ ⟩ = i ℏ | ψ ⟩ ≠ 0. {\displaystyle [{\hat {x}},{\hat {p}}]|\psi \rangle =i\hbar |\psi \rangle \neq 0.} This implies that no quantum state can simultaneously be both

4161-501: The particle between a and b is P ⁡ [ a ≤ X ≤ b ] = ∫ a b | ψ ( x ) | 2 d x   . {\displaystyle \operatorname {P} [a\leq X\leq b]=\int _{a}^{b}|\psi (x)|^{2}\,\mathrm {d} x~.} In the case of the single-mode plane wave, | ψ ( x ) | 2 {\displaystyle |\psi (x)|^{2}}

4234-409: The particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the de Broglie relation p = ħk , where k is the wavenumber . In matrix mechanics , the mathematical formulation of quantum mechanics , any pair of non- commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents

4307-642: The picture where the coherent state is a massive particle in a quantum harmonic oscillator, the position and momentum operators may be expressed in terms of the annihilation operators in the same formulas above and used to calculate the variances, σ x 2 = ℏ 2 m ω , {\displaystyle \sigma _{x}^{2}={\frac {\hbar }{2m\omega }},} σ p 2 = ℏ m ω 2 . {\displaystyle \sigma _{p}^{2}={\frac {\hbar m\omega }{2}}.} Therefore, every coherent state saturates

4380-1679: The position is improved, i.e. reduced σ x , by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σ p . Another way of stating this is that σ x and σ p have an inverse relationship or are at least bounded from below. This is the uncertainty principle, the exact limit of which is the Kennard bound. We are interested in the variances of position and momentum, defined as σ x 2 = ∫ − ∞ ∞ x 2 ⋅ | ψ ( x ) | 2 d x − ( ∫ − ∞ ∞ x ⋅ | ψ ( x ) | 2 d x ) 2 {\displaystyle \sigma _{x}^{2}=\int _{-\infty }^{\infty }x^{2}\cdot |\psi (x)|^{2}\,dx-\left(\int _{-\infty }^{\infty }x\cdot |\psi (x)|^{2}\,dx\right)^{2}} σ p 2 = ∫ − ∞ ∞ p 2 ⋅ | φ ( p ) | 2 d p − ( ∫ − ∞ ∞ p ⋅ | φ ( p ) | 2 d p ) 2   . {\displaystyle \sigma _{p}^{2}=\int _{-\infty }^{\infty }p^{2}\cdot |\varphi (p)|^{2}\,dp-\left(\int _{-\infty }^{\infty }p\cdot |\varphi (p)|^{2}\,dp\right)^{2}~.} Without loss of generality , we will assume that

4453-465: The possible momentum components the particle could have are more widespread. Conversely, the more localized the momentum-space wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly the less localized the position-space wavefunction, so the position coordinates the particle could occupy are more widespread. These wavefunctions are Fourier transforms of each other: mathematically,

SECTION 60

#1732779920626

4526-761: The quantity n 2 π 2 3 − 2 {\textstyle {\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}} is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when n = 1 {\displaystyle n=1} , in which case σ x σ p = ℏ 2 π 2 3 − 2 ≈ 0.568 ℏ > ℏ 2 . {\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {\pi ^{2}}{3}}-2}}\approx 0.568\hbar >{\frac {\hbar }{2}}.} Assume

4599-418: The relations Ω 2 ω 2 + ω 2 Ω 2 ≥ 2 , | cos ⁡ ( 4 ω t ) | ≤ 1 , {\displaystyle {\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\geq 2,\quad |\cos(4\omega t)|\leq 1,} we can conclude

4672-812: The relative contribution of the mode p n to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized. We may take this a step further to the continuum limit , where the wave function is an integral over all possible modes ψ ( x ) = 1 2 π ℏ ∫ − ∞ ∞ φ ( p ) ⋅ e i p x / ℏ d p   , {\displaystyle \psi (x)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\varphi (p)\cdot e^{ipx/\hbar }\,dp~,} with φ ( p ) {\displaystyle \varphi (p)} representing

4745-406: The same time. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of

4818-822: The square root σ x σ p ≥ ℏ 2   . {\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}~.} with equality if and only if p and x are linearly dependent. Note that the only physics involved in this proof was that ψ ( x ) {\displaystyle \psi (x)} and φ ( p ) {\displaystyle \varphi (p)} are wave functions for position and momentum, which are Fourier transforms of each other. A similar result would hold for any pair of conjugate variables. In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators. When considering pairs of observables, an important quantity

4891-460: The standard deviations is therefore σ x σ p = ℏ 2 n 2 π 2 3 − 2 . {\displaystyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}.} For all n = 1 , 2 , 3 , … {\displaystyle n=1,\,2,\,3,\,\ldots } ,

4964-407: The state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. However, the particular eigenstate of the observable A need not be an eigenstate of another observable B : If so, then it does not have a unique associated measurement for it, as the system

5037-450: The uncertainty principle expresses the relationship between conjugate variables in the transform. According to the de Broglie hypothesis , every object in the universe is associated with a wave . Thus every object, from an elementary particle to atoms, molecules and on up to planets and beyond are subject to the uncertainty principle. The time-independent wave function of a single-moded plane wave of wavenumber k 0 or momentum p 0

5110-2026: The variance for momentum can be written as σ p 2 = ∫ − ∞ ∞ | g ~ ( p ) | 2 d p = ∫ − ∞ ∞ | g ( x ) | 2 d x = ⟨ g ∣ g ⟩ . {\displaystyle \sigma _{p}^{2}=\int _{-\infty }^{\infty }|{\tilde {g}}(p)|^{2}\,dp=\int _{-\infty }^{\infty }|g(x)|^{2}\,dx=\langle g\mid g\rangle .} The Cauchy–Schwarz inequality asserts that σ x 2 σ p 2 = ⟨ f ∣ f ⟩ ⋅ ⟨ g ∣ g ⟩ ≥ | ⟨ f ∣ g ⟩ | 2   . {\displaystyle \sigma _{x}^{2}\sigma _{p}^{2}=\langle f\mid f\rangle \cdot \langle g\mid g\rangle \geq |\langle f\mid g\rangle |^{2}~.} The modulus squared of any complex number z can be expressed as | z | 2 = ( Re ( z ) ) 2 + ( Im ( z ) ) 2 ≥ ( Im ( z ) ) 2 = ( z − z ∗ 2 i ) 2 . {\displaystyle |z|^{2}={\Big (}{\text{Re}}(z){\Big )}^{2}+{\Big (}{\text{Im}}(z){\Big )}^{2}\geq {\Big (}{\text{Im}}(z){\Big )}^{2}=\left({\frac {z-z^{\ast }}{2i}}\right)^{2}.} we let z = ⟨ f | g ⟩ {\displaystyle z=\langle f|g\rangle } and z ∗ = ⟨ g ∣ f ⟩ {\displaystyle z^{*}=\langle g\mid f\rangle } and substitute these into

5183-544: The variances may be computed directly, σ x 2 = ℏ m ω ( n + 1 2 ) {\displaystyle \sigma _{x}^{2}={\frac {\hbar }{m\omega }}\left(n+{\frac {1}{2}}\right)} σ p 2 = ℏ m ω ( n + 1 2 ) . {\displaystyle \sigma _{p}^{2}=\hbar m\omega \left(n+{\frac {1}{2}}\right)\,.} The product of these standard deviations

5256-421: The wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle. Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in terms of the creation and annihilation operators : x ^ = ℏ 2 m ω (

5329-1622: The width of the initial state but need not be the same as ω . Through integration over the propagator , we can solve for the full time-dependent solution. After many cancelations, the probability densities reduce to | Ψ ( x , t ) | 2 ∼ N ( x 0 cos ⁡ ( ω t ) , ℏ 2 m Ω ( cos 2 ⁡ ( ω t ) + Ω 2 ω 2 sin 2 ⁡ ( ω t ) ) ) {\displaystyle |\Psi (x,t)|^{2}\sim {\mathcal {N}}\left(x_{0}\cos {(\omega t)},{\frac {\hbar }{2m\Omega }}\left(\cos ^{2}(\omega t)+{\frac {\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega t)}\right)\right)} | Φ ( p , t ) | 2 ∼ N ( − m x 0 ω sin ⁡ ( ω t ) , ℏ m Ω 2 ( cos 2 ⁡ ( ω t ) + ω 2 Ω 2 sin 2 ⁡ ( ω t ) ) ) , {\displaystyle |\Phi (p,t)|^{2}\sim {\mathcal {N}}\left(-mx_{0}\omega \sin(\omega t),{\frac {\hbar m\Omega }{2}}\left(\cos ^{2}{(\omega t)}+{\frac {\omega ^{2}}{\Omega ^{2}}}\sin ^{2}{(\omega t)}\right)\right),} where we have used

#625374