# Quantum Uncertainty Dynamics

Md. Manirul Ali<EMAIL_ADDRESS>Centre for Quantum Science and
Technology, Chennai Institute of Technology, Chennai 600069, India

###### Abstract

Quantum uncertainty relations have deep-rooted significance on the formalism
of quantum mechanics. Heisenberg’s uncertainty relations attracted a renewed
interest for its applications in quantum information science. Robertson
derived a general form of Heisenberg’s uncertainty relations for a pair of
arbitrary observables represented by Hermitian operators. In the present work,
we discover a temporal version of the Heisenberg-Robertson uncertainty
relations for the measurement of two observables at two different times, where
the dynamical uncertainties crucially depend on the time evolution of the
observables. The uncertainties not only depend on the choice of observables,
but they also depend on the times at which the physical observables are
measured. The time correlated two-time commutator dictates the trade-off
between the dynamical uncertainties. We demonstrate the dynamics of these
uncertainty relations for a spin-1/2 system and for a quantum harmonic
oscillator. The temporal uncertainty relations discovered in this work can be
experimentally verified with the present quantum technology.

###### pacs:

03.65.Ta,42.50.-p,42.50.Ar

In classical mechanics, particles can have definite trajectory with precise
position and momentum at any given time. This picture is not valid in quantum
mechanics. Heisenberg’s uncertainty principle gives a restriction on the
accuracy of position and momentum measurements for quantum particles.
Heisenberg’s uncertainty principle Heisenberg ; Kennard  states that if the
momentum of a particle is measured with an uncertainty $\Delta p$ then the
position measurement uncertainty $\Delta x$ is restricted by the relation

$\displaystyle\Delta x\Delta p\geq\frac{\hbar}{2},$ (1)

where $\hbar$ is the reduced Planck constant. Heisenberg in his 1927 paper
Heisenberg  asserted that this relation is a consequence of the quantum
commutation rule for the position and momentum operators
$\left[x,p\right]=i\hbar$. Apart from position and momentum, uncertainty
principle applies to many other pairs of observables including phase and
excitation number of a harmonic oscillator, angle and the orbital angular
momentum of a particle, and components of spin or total angular momentum.
Robertson derived Robertson  a general form of Heisenberg’s uncertainty
relations for a pair of arbitrary observables represented by two Hermitian
operators $A$ and $B$ as follows

$\displaystyle\Delta A\Delta
B\geq\frac{1}{2}\Big{|}\langle[A,B]\rangle\Big{|},$ (2)

which gives a lower bound to the product of variances of the two observables.
The expectation value and the variances are obtained with respect to some
state. The uncertainty relation in terms of sum of variances was also derived
for incompatible observables ArunPati . Quantum uncertainty relations have
deep-rooted significance on the formalism of quantum mechanics Busch .
Uncertainty principle has been the focus due to its applications in quantum
information science RMP17 , specifically in entanglement detection Entang1 ;
Entang2 , security analysis in quantum cryptography Crypto , and determining
quantum nonlocality Nonlocality .

In the present work, we generalize Heisenberg-Robertson uncertainty relation
(2) for the measurement of physical observables $A$ and $B$ at two different
times $t_{1}$ and $t_{2}$. In this situation, we derive the following
uncertainty relation

$\displaystyle\Delta A(t_{1})\Delta
B(t_{2})\geq\frac{1}{2}\Big{|}\langle[A(t_{1}),B(t_{2})]\rangle\Big{|},$ (3)

where the variances $(\Delta A(t_{1}))^{2}=\langle A^{2}(t_{1})\rangle-\langle
A(t_{1})\rangle^{2}$ and $(\Delta B(t_{2}))^{2}=\langle
B^{2}(t_{2})\rangle-\langle B(t_{2})\rangle^{2}$ measure the time-dependent
uncertainties for the observables $A(t_{1})$ and $B(t_{2})$ with respect to
some state $|\Psi\rangle$. The above uncertainty relation (3) is a temporal
version of the Heisenberg-Robertson uncertainty relation (2). The dynamical
uncertainties $\Delta A(t_{1})$ and $\Delta B(t_{2})$ crucially depend on the
dynamics of the observables, where the time dynamics of the commutator
$[A(t_{1}),B(t_{2})]$ determines the trade-off between the measurement
knowledge for the two observables. Next, we derive the dynamical uncertainty
relation (3) when a quantum system evolves under a particular Hamiltonian $H$.
We consider here a time-independent Hamiltonian for simplicity. The time
evolution of the operators are governed by the Heisenberg equation of motion

$\displaystyle\frac{d}{dt}A(t)=\frac{1}{i\hbar}\left[A(t),H\right],$ (4)
$\displaystyle\frac{d}{dt}B(t)=\frac{1}{i\hbar}\left[B(t),H\right],$ (5)

where $A(t)$$=e^{iHt/\hbar}$$A$$e^{-iHt/\hbar}$ and
$B(t)$$=e^{iHt/\hbar}$$B$$e^{-iHt/\hbar}$ are the Heisenberg picture
operators. The operators $A(t)$ and $B(t)$ are Hermitian at any given time. We
consider the measurement of observables $A(t_{1})$ and $B(t_{2})$ at two
different times $t_{1}$ and $t_{2}$. The expectation values of the two
Hermitian operators $A(t_{1})$ and $B(t_{2})$ with respect to a quantum state
$|\Psi\rangle$ are $\langle A(t_{1})\rangle=\langle\Psi|A(t_{1})|\Psi\rangle$
and $\langle B(t_{2})\rangle=\langle\Psi|B(t_{2})|\Psi\rangle$. We define the
fluctuation operators $\delta A(t_{1})$ and $\delta B(t_{2})$ as follows

$\displaystyle\delta A(t_{1})=A(t_{1})-\langle A(t_{1})\rangle,$ (6)
$\displaystyle\delta B(t_{2})=B(t_{2})-\langle B(t_{2})\rangle.$ (7)

Then the following expectation values

$\displaystyle\langle(\delta A(t_{1}))^{2}\rangle=\langle\Psi|(\delta
A(t_{1}))^{2}|\Psi\rangle=(\Delta A(t_{1}))^{2},$ (8)
$\displaystyle\langle(\delta B(t_{2}))^{2}\rangle=\langle\Psi|(\delta
B(t_{2}))^{2}|\Psi\rangle=(\Delta B(t_{2}))^{2},$ (9)

estimate the the mean square deviations or the variances of the two operators.
The action of the fluctuation operators (6) and (7) on the state
$|\Psi\rangle$ are as follows

$\displaystyle|\Psi_{1}\rangle=\delta
A(t_{1})|\Psi\rangle=\left(A(t_{1})-\langle
A(t_{1})\rangle\right)|\Psi\rangle,$ (10)
$\displaystyle|\Psi_{2}\rangle=\delta
B(t_{2})|\Psi\rangle=\left(B(t_{2})-\langle
B(t_{2})\rangle\right)|\Psi\rangle.$ (11)

The inner products of the states $|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$
satisfy the Schwarz inequality given by

$\displaystyle\langle\Psi_{1}|\Psi_{1}\rangle\langle\Psi_{2}|\Psi_{2}\rangle\geq\Big{|}\langle\Psi_{1}|\Psi_{2}\rangle\Big{|}^{2}.$
(12)

Substituting $|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ from Eqs. (10) and (11)
to the above Schwarz inequality (12) we have

$\displaystyle\langle(\delta A(t_{1}))^{2}\rangle~{}\langle(\delta
B(t_{2}))^{2}\rangle\geq\Big{|}\langle\delta A(t_{1})\delta
B(t_{2})\rangle\Big{|}^{2},$ (13)

where all the expectation values in (13) are with respect to the state
$|\Psi\rangle$. The expectation value $\langle\delta A(t_{1})\delta
B(t_{2})\rangle$ on the right of (13) can be expressed as an averaged sum of a
commutator $\langle[A(t_{1}),B(t_{2})]\rangle/2$ and an anticommutator
$\langle\\{\delta A(t_{1}),\delta B(t_{2})\\}\rangle/2$. Note that $\delta
A(t_{1})$ and $\delta B(t_{2})$ are Hermitian as the operators $A(t_{1})$ and
$B(t_{2})$ are both Hermitian. Now the commutator $[A(t_{1}),B(t_{2})]$ is
anti-Hermitian, the expectation value of which is pure imaginary. Whereas the
anticommutator $\\{\delta A(t_{1}),\delta B(t_{2})\\}$ is Hermitian and its
expectation value is real. Hence the squared modulus $\big{|}\langle\delta
A(t_{1})\delta B(t_{2})\rangle\big{|}^{2}$ on the right hand side of (13) can
be written as a sum of two terms

$\displaystyle\frac{1}{4}\Big{|}\langle[A(t_{1}),B(t_{2})]\rangle\Big{|}^{2}+\frac{1}{4}\Big{|}\langle\\{\delta
A(t_{1}),\delta B(t_{2})\\}\rangle\Big{|}^{2},$ (14)

of which the last term is a positive real number. Hence using (8) and (9), the
inequality (13) reduces to

$\displaystyle\langle(\Delta A(t_{1}))^{2}\rangle~{}\langle(\Delta
B(t_{2}))^{2}\rangle\geq\frac{1}{4}\Big{|}\langle[A(t_{1}),B(t_{2})]\rangle\Big{|}^{2}.$
(15)

Taking square root on both sides of (15), we establish the temporal
uncertainty relation (3). Let us now demonstrate the dynamical Heisenberg-
Robertson uncertainty relation (3) for real physical observables that are
measured at two different times. First we consider a spin-1/2 quantum system
evolving under a particular Hamiltonian ${\mathcal{H}}=\omega S_{z}$. We
consider the measurement of spin observables $S_{x}$ and $S_{y}$ at two
different times $t_{1}$ and $t_{2}$. The time evolution of the spin operators
in the Heisenberg picture are given by

$\displaystyle\frac{d}{dt}S_{x}(t)=\frac{1}{i\hbar}\left[S_{x}(t),{\mathcal{H}}\right],$
(16)
$\displaystyle\frac{d}{dt}S_{y}(t)=\frac{1}{i\hbar}\left[S_{y}(t),{\mathcal{H}}\right],$
(17)

where $S_{x}(t)$ and $S_{y}(t)$ are the spin operators in Heisenberg picture
given by

$\displaystyle
S_{x}(t)=e^{i{\mathcal{H}}t/\hbar}~{}S_{x}~{}e^{-i{\mathcal{H}}t/\hbar},$ (18)
$\displaystyle
S_{y}(t)=e^{i{\mathcal{H}}t/\hbar}~{}S_{y}~{}e^{-i{\mathcal{H}}t/\hbar}.$ (19)

The spin operators

$\displaystyle S_{x}=\frac{\hbar}{2}\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right),S_{y}=\frac{\hbar}{2}\left(\begin{array}[]{cc}0&-i\\\
i&0\\\ \end{array}\right),S_{z}=\frac{\hbar}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right)$ (26)

satisfy the commutation relations $[S_{x},S_{y}]=i\hbar S_{z}$,
$[S_{y},S_{z}]$$=i\hbar S_{x}$, and $[S_{z},S_{x}]$$=i\hbar S_{y}$. Then using
the commutation relations $[S_{x},{\mathcal{H}}]$$=-i\hbar\omega S_{y}$ and
$[S_{y},{\mathcal{H}}]$$=i\hbar\omega S_{x}$, one can obtain Heisenberg
equations of motion (16) and (17) for $S_{x}(t)$ and $S_{y}(t)$ as

$\displaystyle\frac{d}{dt}S_{x}(t)=-\omega
S_{y}(t),~{}\mbox{and}~{}\frac{d}{dt}S_{y}(t)=\omega S_{x}(t).$ (27)

Solutions of the Heisenberg equations gives us the time evolution of the spin
operators

$\displaystyle S_{x}(t)=S_{x}\cos(\omega t)-S_{y}\sin(\omega t),$ (28)
$\displaystyle S_{y}(t)=S_{y}\cos(\omega t)+S_{x}\sin(\omega t).$ (29)

Considering the measurement of spin observables $S_{x}$ and $S_{y}$ at two
different times $t_{1}$ and $t_{2}$, we can write down the temporal version of
the Heisenberg-Robertson uncertainty relation (3) as follows

$\displaystyle\Delta S_{x}(t_{1})\Delta
S_{y}(t_{2})\geq\frac{1}{2}\Big{|}\langle[S_{x}(t_{1}),S_{y}(t_{2})]\rangle\Big{|},$
(30)

where the measurement uncertainties

$\displaystyle\Delta S_{x}(t_{1})=\sqrt{\langle
S^{2}_{x}(t_{1})\rangle-\langle S_{x}(t_{1})\rangle^{2}},$ (31)
$\displaystyle\Delta S_{y}(t_{2})=\sqrt{\langle
S^{2}_{y}(t_{2})\rangle-\langle S_{y}(t_{2})\rangle^{2}}$ (32)

are restricted by the commutator

$\displaystyle\left[S_{x}(t_{1}),S_{y}(t_{2})\right]=i\hbar\cos[\omega(t_{2}-t_{1})]S_{z},$
(33)

and the uncertainty relation (30) takes the form

$\displaystyle\Delta S_{x}(t_{1})\Delta
S_{y}(t_{2})\geq\frac{\hbar}{2}\Big{|}\cos[\omega(t_{2}-t_{1})]\langle
S_{z}\rangle\Big{|}.$ (34)

Hence the measurement uncertainties are now correlated in time. The
uncertainties not only depend on the choice of observables, they also depend
on the time at which the physical observables are measured. The commutator
$\left[S_{x}(t_{1}),S_{y}(t_{2})\right]$ dictates the trade-off between the
dynamical uncertainties. Now, under the limiting situation when
$t_{1}$$=t_{2}$$=t$, the two-time uncertainty relation (34) reduces to the
well known equal time uncertainty relation

$\displaystyle\Delta S_{x}(t)\Delta S_{y}(t)\geq\frac{\hbar}{2}\Big{|}\langle
S_{z}\rangle\Big{|}.$ (35)

We can also interchange the sequence of measurements. Suppose we first measure
$S_{y}$ at $t_{1}$ and then measure $S_{x}$ at a later time $t_{2}$. The
uncertainty relation in that case becomes

$\displaystyle\Delta S_{y}(t_{1})\Delta
S_{x}(t_{2})\geq\frac{1}{2}\Big{|}\langle[S_{y}(t_{1}),S_{x}(t_{2})]\rangle\Big{|},$
(36)

where the measurement uncertainties

$\displaystyle\Delta S_{y}(t_{1})=\sqrt{\langle
S^{2}_{y}(t_{1})\rangle-\langle S_{y}(t_{1})\rangle^{2}},$ (37)
$\displaystyle\Delta S_{x}(t_{2})=\sqrt{\langle
S^{2}_{x}(t_{2})\rangle-\langle S_{x}(t_{2})\rangle^{2}}$ (38)

are restricted by the commutator

$\displaystyle\left[S_{y}(t_{1}),S_{x}(t_{2})\right]=-i\hbar\cos[\omega(t_{2}-t_{1})]S_{z},$
(39)

and the uncertainty relation (36) takes the form

$\displaystyle\Delta S_{y}(t_{1})\Delta
S_{x}(t_{2})\geq\frac{\hbar}{2}\Big{|}\cos[\omega(t_{2}-t_{1})]\langle
S_{z}\rangle\Big{|}.$ (40)

Next, we present the uncertainty relation for the same spin observable $S_{x}$
measured at two different times $t_{1}$ and $t_{2}$. Following the general
form of the Heisenberg-Robertson uncertainty relation (3) we have in this case

$\displaystyle\Delta S_{x}(t_{1})\Delta
S_{x}(t_{2})\geq\frac{1}{2}\Big{|}\langle[S_{x}(t_{1}),S_{x}(t_{2})]\rangle\Big{|}.$
(41)

The commutator on the right hand side of (41) can be evaluated using the time-
evolved spin operator (28) as

$\displaystyle\left[S_{x}(t_{1}),S_{x}(t_{2})\right]=-i\hbar\sin[\omega(t_{2}-t_{1})]S_{z},$
(42)

resulting to the following uncertainty relation

$\displaystyle\Delta S_{x}(t_{1})\Delta
S_{x}(t_{2})\geq\frac{\hbar}{2}\Big{|}\sin[\omega(t_{2}-t_{1})]\langle
S_{z}\rangle\Big{|}.$ (43)

If the two successive measurements are very close in time, then
($t_{2}-t_{1}$)$\rightarrow 0$ and $\sin[\omega(t_{2}-t_{1})]$ $\sim$
$\omega(t_{2}-t_{1})$. The dynamical uncertainty relation (43) in this
situation reduces to

$\displaystyle\Delta S_{x}(t_{1})\Delta
S_{x}(t_{2})\geq\frac{\hbar}{2}\Big{|}\omega(t_{2}-t_{1})\langle
S_{z}\rangle\Big{|},$ (44)

reflecting the fact that when one measures the spin observable $S_{x}$
successively at two different times $t_{1}$ and $t_{2}$ with
($t_{2}-t_{1}$)$\rightarrow 0$, the uncertainty of the second measurement
$\Delta S_{x}(t_{2})$ will be very small. Similarly, the uncertainty relation
corresponding to the measurement of spin component $S_{y}$ at two different
times $t_{1}$ and $t_{2}$ is given by

$\displaystyle\Delta S_{y}(t_{1})\Delta
S_{y}(t_{2})\geq\frac{1}{2}\Big{|}\langle[S_{y}(t_{1}),S_{y}(t_{2})]\rangle\Big{|}.$
(45)

The right hand side commutator of (45) can be evaluated again using the time-
evolved spin operator (29) as

$\displaystyle\left[S_{y}(t_{1}),S_{y}(t_{2})\right]=-i\hbar\sin[\omega(t_{2}-t_{1})]S_{z},$
(46)

resulting to the following uncertainty relation

$\displaystyle\Delta S_{y}(t_{1})\Delta
S_{y}(t_{2})\geq\frac{\hbar}{2}\Big{|}\sin[\omega(t_{2}-t_{1})]\langle
S_{z}\rangle\Big{|}.$ (47)

To verify the time-dependent uncertainty relations, one can prepare the
spin-1/2 system in some quantum state

$\displaystyle|\Psi_{s}\rangle=\cos(\theta/2)|\uparrow\rangle+\sin(\theta/2)|\downarrow\rangle,$
(48)

for which the expectation value

$\displaystyle\langle
S_{z}\rangle=\langle\Psi_{s}|S_{z}|\Psi_{s}\rangle=(\hbar/2)\cos(\theta).$
(49)

Using Eqs. (31), (32), (37), and (38) one can obtain the time-dependent
uncertainties quantified by the variances with respect to the state
$|\Psi_{s}\rangle$ as given below

$\displaystyle\Delta
S_{x}(t_{1})=\frac{\hbar}{2}\sqrt{1-\sin^{2}(\theta)\cos^{2}(\omega t_{1})},$
(50) $\displaystyle\Delta
S_{y}(t_{2})=\frac{\hbar}{2}\sqrt{1-\sin^{2}(\theta)\sin^{2}(\omega t_{2})},$
(51) $\displaystyle\Delta
S_{y}(t_{1})=\frac{\hbar}{2}\sqrt{1-\sin^{2}(\theta)\sin^{2}(\omega t_{1})},$
(52) $\displaystyle\Delta
S_{x}(t_{2})=\frac{\hbar}{2}\sqrt{1-\sin^{2}(\theta)\cos^{2}(\omega t_{2})}.$
(53)

Explicit forms of the time dependent quantum uncertainty relations can be
obtained by substituting these expressions of $\langle S_{z}\rangle$, $\Delta
S_{x}(t_{1})$, $\Delta S_{y}(t_{2})$, $\Delta S_{y}(t_{1})$, and $\Delta
S_{x}(t_{2})$ into the equations (34), (40), (43), and (47). The resulting
temporal uncertainty relations can be experimentally verified with the present
superconducting quantum technology SCQ  or with trapped ion qubits Blatt08 ;
Kirchmair09 .

Next, we explore the temporal version of the Heisenberg-Robertson uncertainty
relation (3) for a quantum harmonic oscillator evolving under the Hamiltonian

$\displaystyle{\mathscr{H}}=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}$ (54)

where $X$ and $P$ are the position and momentum operators satisfying the
commutation relation $\left[X,P\right]=i\hbar$. In the Heisenberg picture, the
operators $X$ and $P$ evolves with time according to the equations

$\displaystyle\frac{d}{dt}X(t)=\frac{1}{i\hbar}\left[X(t),{\mathscr{H}}\right],$
(55)
$\displaystyle\frac{d}{dt}P(t)=\frac{1}{i\hbar}\left[P(t),{\mathscr{H}}\right],$
(56)

where $X(t)$ and $P(t)$ are position and momentum operators in the Heisenberg
picture given by

$\displaystyle
X(t)=e^{i{\mathscr{H}}t/\hbar}~{}X~{}e^{-i{\mathscr{H}}t/\hbar},$ (57)
$\displaystyle
P(t)=e^{i{\mathscr{H}}t/\hbar}~{}P~{}e^{-i{\mathscr{H}}t/\hbar}.$ (58)

Using the commutation relations $[X,{\mathscr{H}}]$$=(i\hbar/m)P$ and
$[P,{\mathscr{H}}]$$=(-i\hbar m\omega^{2})X$, one can solve the Heisenberg
equations of motion (55) and (56) to obtain the exact time evolution of the
operators as

$\displaystyle X(t)=X\cos(\omega t)+\frac{1}{m\omega}P\sin(\omega t),$ (59)
$\displaystyle P(t)=P\cos(\omega t)-m\omega X\sin(\omega t).$ (60)

The temporal version of Heisenberg-Robertson uncertainty relation (3) for
measurement of observables $X$ and $P$ at two different times $t_{1}$ and
$t_{2}$ is given by

$\displaystyle\Delta X(t_{1})\Delta
P(t_{2})\geq\frac{1}{2}\Big{|}\langle[X(t_{1}),P(t_{2})]\rangle\Big{|},$ (61)

where the measurement uncertainties $\Delta X(t_{1})$, $\Delta P(t_{2})$, and
the commutator on the right can easily be obtained using the time evolved
operators in (59) and (60)

$\displaystyle\Delta X(t_{1})=\sqrt{\langle X^{2}(t_{1})\rangle-\langle
X(t_{1})\rangle^{2}},$ (62) $\displaystyle\Delta P(t_{2})=\sqrt{\langle
P^{2}(t_{2})\rangle-\langle P(t_{2})\rangle^{2}},$ (63)
$\displaystyle\left[X(t_{1}),P(t_{2})\right]=i\hbar\cos[\omega(t_{2}-t_{1})],$
(64)

that leads the uncertainty relation (61) to the form

$\displaystyle\Delta X(t_{1})\Delta
P(t_{2})\geq\frac{\hbar}{2}\Big{|}\cos[\omega(t_{2}-t_{1})]\Big{|}.$ (65)

The measurement uncertainties are again restricted by the time-correlated
commutator. Under a situation when $t_{1}=t_{2}=t$, the two-time uncertainty
relation (65) reduces to the equal time uncertainty relation

$\displaystyle\Delta X(t)\Delta P(t)\geq\frac{\hbar}{2}.$ (66)

Moreover, we can interchange the order of measurements. We can first measure
momentum $P$ at $t_{1}$ and then measure the position $X$ at a later time
$t_{2}$. Then the form of the uncertainty relation becomes

$\displaystyle\Delta P(t_{1})\Delta
X(t_{2})\geq\frac{1}{2}\Big{|}\langle[P(t_{1}),X(t_{2})]\rangle\Big{|}.$ (67)

where the measurement uncertainties $\Delta P(t_{1})$, $\Delta X(t_{2})$, and
the two-time commutator on the right can be obtained using the time-evolved
position and momentum operators in (59) and (60) as

$\displaystyle\Delta P(t_{1})=\sqrt{\langle P^{2}(t_{1})\rangle-\langle
P(t_{1})\rangle^{2}},$ (68) $\displaystyle\Delta X(t_{2})=\sqrt{\langle
X^{2}(t_{2})\rangle-\langle X(t_{2})\rangle^{2}},$ (69)
$\displaystyle\left[P(t_{1}),X(t_{2})\right]=-i\hbar\cos[\omega(t_{2}-t_{1})],$
(70)

consequently the uncertainty relation (67) takes the form

$\displaystyle\Delta P(t_{1})\Delta
X(t_{2})\geq\frac{\hbar}{2}\Big{|}\cos[\omega(t_{2}-t_{1})]\Big{|}.$ (71)

We now consider the measurement of position $X$ at two different times $t_{1}$
and $t_{2}$. The Heisenberg-Robertson uncertainty relation (3) in this case
becomes

$\displaystyle\Delta X(t_{1})\Delta
X(t_{2})\geq\frac{1}{2}\Big{|}\langle[X(t_{1}),X(t_{2})]\rangle\Big{|}.$ (72)

The two-time commutator on the right hand side of (72) can be evaluated using
the time evolution of the position operator (59) as

$\displaystyle\left[X(t_{1}),X(t_{2})\right]=\frac{i\hbar}{m\omega}\sin[\omega(t_{2}-t_{1})],$
(73)

resulting to the dynamical uncertainty relation

$\displaystyle\Delta X(t_{1})\Delta
X(t_{2})\geq\frac{\hbar}{2m\omega}\Big{|}\sin[\omega(t_{2}-t_{1})]\Big{|}.$
(74)

Consider the two successive measurements are very close in time, then
($t_{2}-t_{1}$)$\rightarrow 0$ and $\sin[\omega(t_{2}-t_{1})]$ $\sim$
$\omega(t_{2}-t_{1})$. The uncertainty relation (74) under that situation
reduces to

$\displaystyle\Delta X(t_{1})\Delta
X(t_{2})\geq\frac{\hbar}{2m}\Big{|}t_{2}-t_{1}\Big{|},$ (75)

which implies that if one measures the observable $X$ successively at two
different times $t_{1}$ and $t_{2}$ with ($t_{2}-t_{1}$)$\rightarrow 0$, the
uncertainty of the second measurement $\Delta X(t_{2})$ will be negligibly
small. Furthermore, the temporal uncertainty relation corresponding to the
measurement of momentum operator $P$ at two different times $t_{1}$ and
$t_{2}$ is given by

$\displaystyle\Delta P(t_{1})\Delta
P(t_{2})\geq\frac{1}{2}\Big{|}\langle[P(t_{1}),P(t_{2})]\rangle\Big{|},$ (76)

where the commutator in the right hand side of (76) can be evaluated again
using the time-evolved momentum operator (60) as

$\displaystyle\left[P(t_{1}),P(t_{2})\right]=i\hbar
m\omega\sin[\omega(t_{2}-t_{1})],$ (77)

that leads to the following uncertainty relation

$\displaystyle\Delta P(t_{1})\Delta P(t_{2})\geq\frac{\hbar
m\omega}{2}\Big{|}\sin[\omega(t_{2}-t_{1})]\Big{|}.$ (78)

For experimental verification of the time-dependent uncertainty relations, one
can prepare the system in a superposition of energy eigenstates given by

$\displaystyle|\Psi_{h}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)$
(79)

where $|0\rangle$ and $|1\rangle$ are the ground state and the first excited
state of the quantum harmonic oscillator. Finally, using the equations (62),
(63), (68), and (69) one can obtain various time-dependent uncertainties
quantified by the variances with respect to this state $|\Psi_{h}\rangle$ as
follows

$\displaystyle\Delta
X(t_{1})=\sqrt{\frac{\hbar}{2m\omega}\big{(}2-\cos^{2}(\omega t_{1})\big{)}},$
(80) $\displaystyle\Delta
P(t_{2})=\sqrt{\frac{m\hbar\omega}{2}\big{(}2-\sin^{2}(\omega t_{2})\big{)}},$
(81) $\displaystyle\Delta
P(t_{1})=\sqrt{\frac{m\hbar\omega}{2}\big{(}2-\sin^{2}(\omega t_{1})\big{)}},$
(82) $\displaystyle\Delta
X(t_{2})=\sqrt{\frac{\hbar}{2m\omega}\big{(}2-\cos^{2}(\omega t_{2})\big{)}}.$
(83)

One can substitute these expressions of $\Delta X(t_{1})$, $\Delta P(t_{2})$,
$\Delta P(t_{1})$, and $\Delta X(t_{2})$ in equations (65), (71), (74), and
(78) to obtain the explicit forms of the time-correlated quantum uncertainty
relations. The resulting dynamical uncertainty relations can be experimentally
verified using for example, nanomechanical oscillator NanoMech ,
superconducting quantum circuit Martinis08 , or quantum optical field
Haroche11 . It will also be interesting to investigate the time-dependent
uncertainty relation (3) for a wide range of quantum mechanical problems.

###### Acknowledgements.

Md. Manirul Ali was supported by the Centre for Quantum Science and
Technology, Chennai Institute of Technology, India, vide funding number
CIT/CQST/2021/RD-007.

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