Bra-ket notation in linear algebra
5.1 Uncertainty relations
The statistical interpretation of the wave function naturally implies uncertainty in an observable Aoif the wave function is not an eigenstate of the hermitian operator A that corresponds to Ao. Suppose that A has eigenvalues an,
Aj ni D anj ni; h mj ni D ımn: Substitution of the expansion
j i DX
n
j nih nj i
© Springer International Publishing Switzerland 2016
R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, DOI 10.1007/978-3-319-25675-7_5
85
86 5 Formal Developments
into the formula for the expectation value hAi D h jAj i in the state j i yields hAi DX
n
anjh nj ij2;
i.e.jh nj ij2is a probability that the value anfor the observable Aowill be observed if the system is in the state j i.
If the distributionjh nj ij2 is strongly concentrated around a particular index
`, then it is very likely that a measurement of Ao will find the value a`with very little uncertainty. However, if the probability distributionjh nj ij2 covers a broad range of indices or has maxima e.g. for two separated indices, then there will be high uncertainty of the value of the observable Ao, and observation of Aofor many copies of the system in the state j i will yield a large scatter of observed values.
If A has a continuous spectrum of eigenvalues, e.g. if A D x is the operator for the location x of a particle in one dimension, thenjhxj ij2dx is the probability to find the system with a value of x in the intervalŒx; x C dx.
Heisenberg found in 1927 an intuitive estimate for the minimal product of uncertaintiesx and p in location and momentum of a particle1. His arguments were easily made rigorous and generalized to other pairs of observables using the statistical formalism of quantum mechanics.
Suppose that two observables Ao and Bo are represented by the two hermitian operators A and B. The expectation value of the observable Aoin a state j i is
hAi D h jAj i;
and the uncertaintyA of the expectation value hAi is defined through
A2D h.A hAi/2i D hA2i hAi2:
Minimal values of the uncertainty A with which the observable Ao can be measured are directly related to the commutator of the operator A with other operators. The commutator of the two operators A and B is defined through
ŒA; Bj i ABj i BAj i;
where ABj i is the action of the operator B on the state j i followed by the action of the operator A on the new state j 0i D Bj i.
The commutator of two hermitian operators yields a new hermitian operator C, ŒA; B D iC;
and it is easy to show that the magnitude of the expectation value hCi yields a lower bound on the product of uncertaintiesA B,
AB 1
2jhCij: (5.1)
1W. Heisenberg, Z. Phys. 43, 172 (1927).
For the proof of this relation, we use a real parameter. The function 0 f ./ D h.A hAi iB C ihBi/ .A hAi C iB ihBi/i
D A2 hCi C 2B2 (5.2)
has minimal value for
D hCi 2B2 and substitution into (5.2) yields
0 A2 hCi2 4B2: This implies the result (5.1).
For the inequality in equation (5.2) note that
h j .A hAi iB C ihBi/ .A hAi C iB ihBi/ j i D j.A hAi C iB ihBi/ j ij2:
Equation (5.1) implies for the operators x and p for location and momentum of a particle Heisenberg’s uncertainty relation
xp „
2; (5.3)
or in tensorial form for the three-dimensional operators x and p
x ˝ p „ 21
If the state j i should satisfy the uncertainty relation for A B with the equality sign (minimal product of uncertainties), then we must have
0 D h j
.A hAi/2 hCi2
4B4.B hBi/2
j i
D h j
A hAi ihCiB hBi 2B2
A hAi C ihCiB hBi 2B2
j i
Dˇˇ ˇˇ
A hAi C ihCiB hBi 2B2
j iˇˇ
ˇˇ2;
88 5 Formal Developments where in the second equation the commutatorŒA; B D iC has been used. This is equivalent to
A hAi C ihCiB hBi 2B2
j i D 0:
In particular we have minimalpx if and only if h
p p0 i„x x0 2x2
ij i D 0: (5.4)
This implies in the x-representation
d dx ip0
„ Cx x0 2x2
hxj i D 0
and yields up to an arbitrary constant phase factor the Gaussian wave packet (3.39)
hxj i D 1
.2x2/1=4exp
.x x0/2 4x2 C ip0x
„ ip0x0 2„
: (5.5)
Equation (5.4) is in p-representation
2x2.p p0/ C i„x0C „2 d dp
hpj i D 0;
and this yields (again up to an arbitrary constant phase factor)
hpj i D
2x2
„2
14 exp
x2
„2 .p p0/2 ipx0
„ C ip0x0 2„
; (5.6)
which explicitly confirms
p D „ 2x and corresponds to the wave packet (3.40).
For comparison of the solutions, we note that hpj i D p1
2„
Z 1
1dx exp
ipx
„
hxj i
D p1 2„
1 .2x2/1=4
Z 1
1dx exp
"
14x2
x x0C i 2x2
„ .p p0/
2#
exp
x2
„2 .p p0/2 i .p p0/x0
„
D
2x2
„2
14 exp
x2
„2 .p p0/2 ipx0
„ C ip0x0 2„
:
The phase factor exp.ip0x0=2„/ was included in in (5.5) to ensure that (5.5) and (5.6) are also related through direct Fourier transformations. Otherwise there would have been a mismatch in a phase / p0x0=2„.
We have seen in equation (3.43) that the widthx2will remain minimal only for a certain moment in time if a Gaussian packet follows a free evolution idj .t/i=dt / p2j .t/i, while the uncertainty p2 in momentum remains constant. Therefore a freely evolving Gaussian packet will satisfy the minimal conditionxp D „=2 only for a moment in time.
A Gaussian wave packet following an evolution idj .t/i=dt / x2j .t/i would have constantx2, butp2would have the minimal possible value „=.2x/ only for a moment in time. Such a hypothetical quantum system would correspond to an oscillator without kinetic energy, and it could move uniformly along the p axis.
In Chapter6we will find that a harmonic oscillator evolution for Gaussian wave packets, idj .t/i=dt D .˛p2C ˇx2/j .t/i, yields constant widths both in x and in p direction, and the minimal uncertainty conditionxp D „=2 will be satisfied at all times.
There is also an uncertainty relation between energy and time, which is not as strict as the relations (5.1, 5.3), and cannot be proven by the same rigorous mathematical methods. The relation involves the minimal time windowt which is required to observe a system with energy uncertainty E. Smaller energy uncertainty requires a longer observation window, or a longer time to form the system,
tE & O.„/: (5.7)
This order of magnitude estimate is often written astE & „=2 for symmetry with the Heisenberg uncertainty relation (5.3), but it should not be mistaken to indicate a strict lower bound as in equation (5.3).
Equation (5.7) cannot be derived in the same way as equation (5.3) because time is not an observable, but a parameter in quantum mechanics. Therefore there is no related expectation value, nor is there any corresponding definition of t as the variance of an expectation value.
There exist a few simple heuristic derivations to motivate equation (5.7) from equation (5.3), but we will find the best justification for (5.7) in the equations of time-dependent perturbation theory in Section13.8.
90 5 Formal Developments