In the last post, we deal with the rotations in quantum mechanics. In the post, we focus on yet another translation, not in space, but in time. From time-evolution of quantum mechanical states, we will meet the SchrΓΆdinger equation and the Hamiltonian, the operator corresponding to the total energy of the system.
The Time-Evolution Operator and the SchrΓΆdinger Equation
Consider a time-dependent state
β£ Ο ( t ) β© |\psi(t)\rangle β£ Ο ( t )β© , weβd like a time-dependent operator U ^ ( t ) \hat U(t) U ^ ( t ) , such that:
U ^ ( t ) β£ Ο ( 0 ) β© = β£ Ο ( t ) β© \hat U (t) |\psi(0)\rangle = |\psi(t)\rangle U ^ ( t ) β£ Ο ( 0 )β© = β£ Ο ( t )β©
What properties should this time-evolution operator have? It turns out it should be very similar to a rotational operator R ^ ( d Ο ) \hat R(d\psi) R ^ ( d Ο ) :
It should act like the identity operator when t = 0 t=0 t = 0 : U ^ ( t = 0 ) = 1 \hat U(t = 0) = \mathbf 1 U ^ ( t = 0 ) = 1
It should be dimensionless.
It should be unitary, a.k.a norm-preserving, i.e.
β¨ Ο ( t ) β£ Ο ( t ) β© = β¨ Ο ( 0 ) β£ U ^ β ( t ) U ^ ( t ) β£ Ο ( 0 ) β© = β¨ Ο ( 0 ) β£ Ο ( 0 ) β© βΊ U β ( t ) U ^ = 1 \langle\psi(t)|\psi(t)\rangle = \langle\psi(0)|\hat U^\dagger (t) \hat U (t)|\psi(0)\rangle = \langle\psi(0)|\psi(0)\rangle \Longleftrightarrow U^\dagger (t) \hat U = \pmb 1 β¨ Ο ( t ) β£ Ο ( t )β© = β¨ Ο ( 0 ) β£ U ^ β ( t ) U ^ ( t ) β£ Ο ( 0 )β© = β¨ Ο ( 0 ) β£ Ο ( 0 )β© βΊ U β ( t ) U ^ = 1 1
It should be reasonable to motivate that the time-evolution operator have a similar form as the rotational operator. And just like the rotational operator has a generator of rotation, spin operator, the time-evolution operator also has its generator. We assert thatβand it will be apparent that this is equivalent to the 5th postulate, the time-dependent SchrΓΆdinger equationβthat:
U ^ ( d t ) = 1 β i β H ^ d t \hat U(dt) = 1 - \frac{i}{\hbar} \hat H dt U ^ ( d t ) = 1 β β i β H ^ d t
where H ^ \hat H H ^ is called the Hamiltonian , the generator of time translation.
Now just like what we did for the rotational operator in the second post, we can obtain a closed form solution for the time-evolution operator by taking a series of infinitesimal time translations. Assuming the Hamiltonian is time-independent:
U ^ ( t ) = lim β‘ N β β ( 1 β i β H ^ t N ) N = e β i H ^ t / β \hat U(t) = \lim_{N \to\infty} (1 - \frac{i}{\hbar} \hat H \frac{t}{N})^N = e^{-i\hat H t/\hbar} U ^ ( t ) = N β β lim β ( 1 β β i β H ^ N t β ) N = e β i H ^ t /β
The time-evolution operator also satisfies a first-order differential equation w.r.t time:
U ^ ( t + d t ) = U ^ ( d t ) U ^ ( t ) = ( 1 β i β H ^ d t ) U ^ ( t ) \hat U(t+dt) = \hat U(dt) \hat U(t) = (1-\frac{i}{\hbar} \hat H dt) \hat U(t) U ^ ( t + d t ) = U ^ ( d t ) U ^ ( t ) = ( 1 β β i β H ^ d t ) U ^ ( t )
Then,
U ^ ( t + d t ) β U ^ ( t ) = β ( i β H ^ d t ) U ^ ( t ) \hat U(t+dt) -\hat U(t) = -(\frac{i}{\hbar} \hat H dt) \hat U(t) U ^ ( t + d t ) β U ^ ( t ) = β ( β i β H ^ d t ) U ^ ( t )
And take the infinitesimal limit on time,
i β d d t U ^ = H ^ U ^ ( t ) i\hbar \frac{d}{dt}\hat U = \hat H \hat U(t) i β d t d β U ^ = H ^ U ^ ( t )
Finally, apply the operator equation on the initial state
β£ Ο ( 0 ) β© |\psi(0)\rangle β£ Ο ( 0 )β© :
i β d d t β£ Ο ( t ) β© = H ^ β£ Ο ( t ) β© i\hbar \frac{d}{dt} |\psi(t)\rangle = \hat H |\psi(t)\rangle i β d t d β β£ Ο ( t )β© = H ^ β£ Ο ( t )β©
There we have the time-dependent SchrΓΆdinger equation .
The Hamiltonian
Letβs come back to the generator of time translation, the Hamiltonian. Like the generator of rotation, the Hamiltonian is a Hermitian operator. Itβs reasonable to postulate that it corresponds to some measurable quantity. Letβs first consider its unit, [ H ^ ] = [ β / t ] [\hat H] = [\hbar/t] [ H ^ ] = [ β/ t ] , which is the unit of energy. Moreover, assuming Hamiltonian is time-independent, its expectation value is time independent.
To see this, we first Taylor-expand the time-evolution operator:
U ^ ( t ) = e β i H ^ t / β = 1 β i H ^ t β + 1 2 ! ( β i H ^ t β ) 2 + β― \hat U(t) = e^{-i\hat H t/\hbar} = 1 - \frac{i\hat H t}{\hbar} + \frac{1}{2!}(-\frac{i\hat H t}{\hbar})^2 + \cdots U ^ ( t ) = e β i H ^ t /β = 1 β β i H ^ t β + 2 ! 1 β ( β β i H ^ t β ) 2 + β―
The expansion shows that H ^ \hat H H ^ commutes with U ^ \hat U U ^ , since U ^ \hat U U ^ is the sum of powers of H ^ \hat H H ^ . Therefore,
β¨ Ο ( t ) β£ H ^ β£ Ο ( t ) β© = β¨ Ο ( 0 ) β£ U ^ β H ^ U ^ β£ Ο ( 0 ) β© = β¨ Ο ( 0 ) β£ U ^ β U ^ H ^ β£ Ο ( 0 ) β© = β¨ Ο ( 0 ) β£ H ^ β£ Ο ( 0 ) β© \langle\psi(t)|\hat H | \psi(t)\rangle = \langle\psi(0)|\hat U^\dagger \hat H \hat U| \psi(0)\rangle = \langle\psi(0)|\hat U^\dagger \hat U \hat H | \psi(0)\rangle = \langle\psi(0)|\hat H | \psi(0)\rangle β¨ Ο ( t ) β£ H ^ β£ Ο ( t )β© = β¨ Ο ( 0 ) β£ U ^ β H ^ U ^ β£ Ο ( 0 )β© = β¨ Ο ( 0 ) β£ U ^ β U ^ H ^ β£ Ο ( 0 )β© = β¨ Ο ( 0 ) β£ H ^ β£ Ο ( 0 )β©
These results suggest, with our classical intuition, the Hamiltonian is the energy operator. The eigenstates of the Hamiltonian is the energy eigenstates
β£ E β© |E\rangle β£ E β© , and the eigenvalues are measurable energy E E E of the system:
H ^ β£ E β© = E β£ E β© \hat H |E\rangle = E |E\rangle H ^ β£ E β© = E β£ E β©
The time-evolution of an energy eigenstate is especially convenient. Again, we use the Taylor-expansion of the time-evolution operator,
U ^ ( t ) β£ E β© = e β i H ^ t / β β£ E β© = [ 1 β i H ^ t β + 1 2 ! ( β i H ^ t β ) 2 + β― ] β£ E β© = [ 1 β i E t β + 1 2 ! ( β i E t β ) 2 + β― ] β£ E β© = e β i E t / β \begin{aligned}
\hat U(t) |E\rangle &= e^{-i\hat H t/\hbar} |E\rangle \\
&= \Big[1 - \frac{i\hat H t}{\hbar} + \frac{1}{2!}(-\frac{i\hat H t}{\hbar})^2 + \cdots \Big] |E\rangle \\
&= \Big[1 - \frac{iE t}{\hbar} + \frac{1}{2!}(-\frac{iE t}{\hbar})^2 + \cdots \Big] |E\rangle \\
&= e^{-i E t/\hbar}
\end{aligned} U ^ ( t ) β£ E β© β = e β i H ^ t /β β£ E β© = [ 1 β β i H ^ t β + 2 ! 1 β ( β β i H ^ t β ) 2 + β― ] β£ E β© = [ 1 β β i Et β + 2 ! 1 β ( β β i Et β ) 2 + β― ] β£ E β© = e β i Et /β β
Time Dependence of Expectation
With the SchrΓΆdinger equation, we can also find the expectation of a measurement.
d d t β¨ A β© = d d t β¨ Ο ( t ) β£ A ^ β£ Ο ( t ) β© = ( d d t β¨ Ο ( t ) β£ A ^ β£ Ο ( t ) β© ) + β¨ Ο ( t ) β£ A ^ ( d d t β£ Ο ( t ) β© ) + β¨ Ο ( t ) β£ β A ^ β t β£ Ο ( t ) β© = ( 1 β i β β¨ Ο ( t ) β£ H ^ ) A ^ β£ Ο ( t ) β© ) + β¨ Ο ( t ) β£ A ^ ( 1 i β H ^ β£ Ο ( t ) β© ) + β¨ Ο ( t ) β£ β A ^ β t β£ Ο ( t ) β© = i β β¨ Ο ( t ) β£ [ H ^ , A ^ ] β£ Ο ( t ) β© + β¨ Ο ( t ) β£ β A ^ β t β£ Ο ( t ) β© \begin{aligned}
\frac{d}{dt} \langle A\rangle
&= \frac{d}{dt}\langle\psi(t)| \hat A |\psi(t)\rangle \\
&= \Big(\frac{d}{dt}\langle\psi(t)| \hat A |\psi(t)\rangle\Big) + \langle\psi(t)| \hat A \Big(\frac{d}{dt}|\psi(t)\rangle\Big) + \langle\psi(t)| \frac{\partial \hat A}{\partial t}|\psi(t)\rangle \\
&= \Big(\frac{1}{-i\hbar}\langle\psi(t)| \hat H) \hat A |\psi(t)\rangle\Big) + \langle\psi(t)| \hat A \Big(\frac{1}{i\hbar}\hat H|\psi(t)\rangle\Big) + \langle\psi(t)| \frac{\partial \hat A}{\partial t}|\psi(t)\rangle \\
&= \frac{i}{\hbar} \langle\psi(t)| \Big[ \hat H, \hat A \Big] |\psi(t)\rangle + \langle\psi(t)| \frac{\partial \hat A}{\partial t}|\psi(t)\rangle
\end{aligned} d t d β β¨ A β© β = d t d β β¨ Ο ( t ) β£ A ^ β£ Ο ( t )β© = ( d t d β β¨ Ο ( t ) β£ A ^ β£ Ο ( t )β© ) + β¨ Ο ( t ) β£ A ^ ( d t d β β£ Ο ( t )β© ) + β¨ Ο ( t ) β£ β t β A ^ β β£ Ο ( t )β© = ( β i β 1 β β¨ Ο ( t ) β£ H ^ ) A ^ β£ Ο ( t )β© ) + β¨ Ο ( t ) β£ A ^ ( i β 1 β H ^ β£ Ο ( t )β© ) + β¨ Ο ( t ) β£ β t β A ^ β β£ Ο ( t )β© = β i β β¨ Ο ( t ) β£ [ H ^ , A ^ ] β£ Ο ( t )β© + β¨ Ο ( t ) β£ β t β A ^ β β£ Ο ( t )β© β
where in the second to last step we apply the SchrΓΆdinger equation and its adjoint complement
β i β d d t β¨ Ο ( t ) β£ = β¨ Ο ( t ) β£ H ^ -i\hbar \frac{d}{dt} \langle\psi(t)| = \langle\psi(t)| \hat H β i β d t d β β¨ Ο ( t ) β£ = β¨ Ο ( t ) β£ H ^ .
This relationship tells us that an expectation value can be time-dependent if the operator does not commute with the Hamiltonian or if the operator time-dependent.
Precession of Spin-1/2 Particles in a B-field
Indeed, we will have an example of a time-dependent operator. In fact a time-dependent Hamiltonian, which will result in a time-dependent expectation value for the energy of the system.
Suppose we have a homogeneous magnetic field Β± B = B 0 k ^ \pm B = B_0 \hat k Β± B = B 0 β k ^ . Assume we have an electron with no kinetic energy in the field. In the classical mechanics, the total energy will be its potential energy,
E = β ΞΌ β
B = β g q 2 m e c S β
B = g e 2 m e c B 0 S z = Ο 0 S z where Ο 0 = g e 2 m e c E = - \pmb \mu \cdot \pmb B = - \frac{gq}{2m_ec} \pmb S \cdot \pmb B = \frac{ge}{2m_ec} B_0 S_z = \omega_0 S_z \text{ where } \omega_0 = \frac{ge}{2m_ec} E = β ΞΌ ΞΌ β
B B = β 2 m e β c g q β S S β
B B = 2 m e β c g e β B 0 β S z β = Ο 0 β S z β where Ο 0 β = 2 m e β c g e β
But this is erroneous, since we cannot even comprehend spin S \pmb S S S in the classical mechanics. In the quantum mechanics, we need to express the Hamiltonian with the spin operator:
H ^ = Ο 0 S z ^ \hat H = \omega_0 \hat{S_z} H ^ = Ο 0 β S z β ^ β
The hamiltonian has the same eigenstates as
S z ^ \hat{S_z} S z β ^ β , β£ Β± z β© |\pm z\rangle β£ Β± z β© . The energy eigenvalues can be calculated:
H ^ β£ Β± z β© = Ο 0 S z ^ β£ Β± z β© = Β± β Ο 0 2 β£ Β± z β© = E Β± β£ Β± z β© \hat H |\pm z\rangle = \omega_0 \hat{S_z} |\pm z\rangle = \pm \frac{\hbar \omega_0}{2} |\pm z\rangle = E_\pm |\pm z\rangle H ^ β£ Β± z β© = Ο 0 β S z β ^ β β£ Β± z β© = Β± 2 β Ο 0 β β β£ Β± z β© = E Β± β β£ Β± z β©
So the energy eigenvalues are E Β± = Β± β Ο 0 2 E_\pm = \pm \frac{\hbar \omega_0}{2} E Β± β = Β± 2 β Ο 0 β β for
β£ Β± z β© |\pm z\rangle β£ Β± z β© .
What will happen as time evolves? At time t t t , the time-evolution operator is:
U ^ ( t ) = e β i H ^ t / β = e β i Ο 0 S z ^ t / β = R ^ z ( Ο 0 t ) \hat U(t) = e^{-i\hat H t/\hbar} = e^{-i \omega_0 \hat{S_z} t/\hbar} = \hat R_z(\omega_0 t) U ^ ( t ) = e β i H ^ t /β = e β i Ο 0 β S z β ^ β t /β = R ^ z β ( Ο 0 β t )
The time-evolution operator is the rotation operator about the z axis! This means that the electronβs spin will precess about the z-axis with the period T = 2 Ο / Ο 0 T=2\pi/\omega_0 T = 2 Ο / Ο 0 β .
For an electron originally in the states
β£ + x β© = 1 2 β£ + z β© + 1 2 β£ β z β© |+ x\rangle = \frac{1}{\sqrt{2}} |+ z\rangle + \frac{1}{\sqrt{2}} |- z\rangle β£ + x β© = 2 β 1 β β£ + z β© + 2 β 1 β β£ β z β© , we can calculate the expectation value of S z S_z S z β at time t t t ,
β¨ S z β© = β¨ + x β£ U ^ β ( t ) S z ^ U ^ ( t ) β£ + x β© = ( 1 2 β¨ + z β£ + 1 2 β¨ β z β£ ) e i Ο 0 S z ^ t / β S z ^ e β i Ο 0 S z ^ t / β ( 1 2 β£ + z β© + 1 2 β£ β z β© ) = 1 2 ( e i Ο 0 t / 2 β¨ + z β£ + e β i Ο 0 t / 2 β¨ β z β£ ) S z ^ 1 2 ( e β i Ο 0 t / 2 β£ + z β© + e i Ο 0 t / 2 β£ β z β© ) = 1 2 ( e i Ο 0 t / 2 e β i Ο 0 t / 2 ) β 2 ( 1 0 0 β 1 ) ( e β i Ο 0 t / 2 e i Ο 0 t / 2 ) = β 2 1 2 ( 1 β 1 ) = 0 \begin{aligned}
\langle S_z\rangle &= \langle+x|\hat U^\dagger(t) \hat{S_z} \hat U(t)|+x\rangle \\
&= \Big(\frac{1}{\sqrt{2}} \langle+z|+\frac{1}{\sqrt{2}} \langle-z|\Big) e^{i \omega_0 \hat{S_z} t/\hbar} \hat{S_z} e^{-i \omega_0 \hat{S_z} t/\hbar} \Big( \frac{1}{\sqrt{2}} |+ z\rangle + \frac{1}{\sqrt{2}} |- z\rangle\Big) \\
&= \frac{1}{\sqrt{2}} \Big( e^{i \omega_0 t/2} \langle+z| + e^{- i \omega_0 t/2} \langle-z| \Big) \hat{S_z} \frac{1}{\sqrt{2}} \Big( e^{-i \omega_0 t/2} |+z\rangle + e^{i \omega_0 t/2} |-z\rangle \Big) \\
&= \frac{1}{2} \begin{pmatrix} e^{i \omega_0 t/2} & e^{-i \omega_0 t/2} \end{pmatrix} \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} e^{-i \omega_0 t/2} \\ e^{i \omega_0 t/2} \end{pmatrix} \\
&= \frac{\hbar}{2} \frac{1}{2} (1 -1) = 0
\end{aligned} β¨ S z β β© β = β¨ + x β£ U ^ β ( t ) S z β ^ β U ^ ( t ) β£ + x β© = ( 2 β 1 β β¨ + z β£ + 2 β 1 β β¨ β z β£ ) e i Ο 0 β S z β ^ β t /β S z β ^ β e β i Ο 0 β S z β ^ β t /β ( 2 β 1 β β£ + z β© + 2 β 1 β β£ β z β© ) = 2 β 1 β ( e i Ο 0 β t /2 β¨ + z β£ + e β i Ο 0 β t /2 β¨ β z β£ ) S z β ^ β 2 β 1 β ( e β i Ο 0 β t /2 β£ + z β© + e i Ο 0 β t /2 β£ β z β© ) = 2 1 β ( e i Ο 0 β t /2 β e β i Ο 0 β t /2 β ) 2 β β ( 1 0 β 0 β 1 β ) ( e β i Ο 0 β t /2 e i Ο 0 β t /2 β ) = 2 β β 2 1 β ( 1 β 1 ) = 0 β
The result is as expected. Intuitively, we can think of the spin precessing about the z-axis in the x-y plane, and the measurement of S z S_z S z β should be time-independent and 0 in expectation.
Similarly, we can calculate β¨ S x β© \langle S_x\rangle β¨ S x β β© ,
β¨ S x β© = β¨ + x β£ U ^ β ( t ) S x ^ U ^ ( t ) β£ + x β© = ( 1 2 β¨ + z β£ + 1 2 β¨ β z β£ ) e i Ο 0 S z ^ t / β S x ^ e β i Ο 0 S z ^ t / β ( 1 2 β£ + z β© + 1 2 β£ β z β© ) = 1 2 ( e i Ο 0 t / 2 β¨ + z β£ + e β i Ο 0 t / 2 β¨ β z β£ ) S x ^ 1 2 ( e β i Ο 0 t / 2 β£ + z β© + e i Ο 0 t / 2 β£ β z β© ) = 1 2 ( e i Ο 0 t / 2 e β i Ο 0 t / 2 ) β 2 ( 0 1 1 0 ) ( e β i Ο 0 t / 2 e i Ο 0 t / 2 ) = β 2 1 2 ( e i Ο 0 t + e β i Ο 0 t ) = β 2 c o s ( Ο 0 t ) \begin{aligned}
\langle S_x\rangle
&= \langle+x|\hat U^\dagger(t) \hat{S_x} \hat U(t)|+x\rangle \\
&= \Big(\frac{1}{\sqrt{2}} \langle+z|+\frac{1}{\sqrt{2}} \langle-z|\Big) e^{i \omega_0 \hat{S_z} t/\hbar} \hat{S_x} e^{-i \omega_0 \hat{S_z} t/\hbar} \Big( \frac{1}{\sqrt{2}} |+ z\rangle + \frac{1}{\sqrt{2}} |- z\rangle\Big) \\
&= \frac{1}{\sqrt{2}} \Big( e^{i \omega_0 t/2} \langle+z| + e^{- i \omega_0 t/2} \langle-z| \Big) \hat{S_x} \frac{1}{\sqrt{2}} \Big( e^{-i \omega_0 t/2} |+z\rangle + e^{i \omega_0 t/2} |-z\rangle \Big) \\
&= \frac{1}{2} \begin{pmatrix} e^{i \omega_0 t/2} & e^{-i \omega_0 t/2} \end{pmatrix} \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} e^{-i \omega_0 t/2} \\ e^{i \omega_0 t/2} \end{pmatrix} \\
&= \frac{\hbar}{2} \frac{1}{2} ( e^{i \omega_0t} + e^{-i \omega_0t}) = \frac{\hbar}{2} cos(\omega_0 t)
\end{aligned} β¨ S x β β© β = β¨ + x β£ U ^ β ( t ) S x β ^ β U ^ ( t ) β£ + x β© = ( 2 β 1 β β¨ + z β£ + 2 β 1 β β¨ β z β£ ) e i Ο 0 β S z β ^ β t /β S x β ^ β e β i Ο 0 β S z β ^ β t /β ( 2 β 1 β β£ + z β© + 2 β 1 β β£ β z β© ) = 2 β 1 β ( e i Ο 0 β t /2 β¨ + z β£ + e β i Ο 0 β t /2 β¨ β z β£ ) S x β ^ β 2 β 1 β ( e β i Ο 0 β t /2 β£ + z β© + e i Ο 0 β t /2 β£ β z β© ) = 2 1 β ( e i Ο 0 β t /2 β e β i Ο 0 β t /2 β ) 2 β β ( 0 1 β 1 0 β ) ( e β i Ο 0 β t /2 e i Ο 0 β t /2 β ) = 2 β β 2 1 β ( e i Ο 0 β t + e β i Ο 0 β t ) = 2 β β cos ( Ο 0 β t ) β
which has a clear time-dependence.
Energy-Time Uncertainty
From the last post, we know from the the generalized uncertainty principle that
Ξ A Ξ B β₯ 1 2 β£ β¨ [ A ^ , B ^ ] β© β£ \Delta A \Delta B \geq \frac {1}{2} |\langle \Big[ \hat A , \hat B \Big] \rangle| Ξ A Ξ B β₯ 2 1 β β£ β¨ [ A ^ , B ^ ] β© β£
For a time-independent operator A ^ \hat A A ^ , we have
Ξ A Ξ E β₯ 1 2 β¨ Ο β£ [ A ^ , H ^ ] β£ Ο β© \Delta A \Delta E \geq \frac{1}{2} \langle\psi|\Big[\hat A, \hat H\Big]|\psi\rangle Ξ A Ξ E β₯ 2 1 β β¨ Ο β£ [ A ^ , H ^ ] β£ Ο β©
and from the time dependence of expectation,
d β¨ A β© d t = i β β¨ Ο ( t ) β£ [ H ^ , A ^ ] β£ Ο ( t ) β© \frac{d\langle A\rangle}{dt} = \frac{i}{\hbar} \langle\psi(t)| \Big[\hat H, \hat A \Big] |\psi(t)\rangle d t d β¨ A β© β = β i β β¨ Ο ( t ) β£ [ H ^ , A ^ ] β£ Ο ( t )β©
These two equations combined tell us that
Ξ A Ξ E β₯ β 2 β£ d β¨ A β© d t β£ Ξ E Ξ A β£ d β¨ A β© d t β£ β₯ β 2 Ξ E Ξ t β₯ β 2 \begin{aligned}
\Delta A \Delta E &\geq \frac{\hbar}{2} |\frac{d\langle A\rangle}{dt}| \\
\Delta E \frac{\Delta A}{|\frac{d\langle A\rangle}{dt}|} &\geq \frac{\hbar}{2} \\
\Delta E \Delta t &\geq \frac{\hbar}{2}
\end{aligned} Ξ A Ξ E Ξ E β£ d t d β¨ A β© β β£ Ξ A β Ξ E Ξ t β β₯ 2 β β β£ d t d β¨ A β© β β£ β₯ 2 β β β₯ 2 β β β
where
Ξ t = Ξ A β£ d β¨ A β© d t β£ \Delta t = \frac{\Delta A}{|\frac{d\langle A\rangle}{dt}|} Ξ t = β£ d t d β¨ A β© β β£ Ξ A β is the time for β¨ A β© \langle A\rangle β¨ A β© to change by Ξ A \Delta A Ξ A .
References
[1] Zajc, William. PHYS GU4021-4022 Quantum Mechanics I-II, Department of Physics, Columbia University. 2019-2020.
[2] Townsend, John. A Modern Approach to Quantum Mechanics. 2nd Edition.