Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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A^ in equation (3.72) equal first to the position
variable x, then the variable y, and finally the variable z, we can show that
98 General principles of quantum theory
m
dhxi
dt
ˆ hpxi, m dhyi
dt
ˆ hpyi, m dhzi
dt
ˆ hpzi
or, in vector notation
m
dhri
dt
ˆ hpi
which is one of the Ehrenfest theorems discussed in Section 2.3. The other
Ehrenfest theorem,
dhpi
dt
ˆ ÿh=V (r)i
may be obtained from equation (3.72) by setting A^ successively equal to p^x,
p^y, and p^z.
3.11 Heisenberg uncertainty principle
We have shown in Section 3.5 that commuting hermitian operators have
simultaneous eigenfunctions and, therefore, that the physical quantities asso-
ciated with those operators can be observed simultaneously. On the other hand,
if the hermitian operators A^ and B^ do not commute, then the physical
observables A and B cannot both be precisely determined at the same time. We
begin by demonstrating this conclusion.
Suppose that A^ and B^ do not commute. Let Æi and âi be the eigenvalues of A^
and B^, respectively, with corresponding eigenstates jÆii and jâii
A^jÆii ˆ ÆijÆii (3:73a)
B^jâii ˆ âijâii (3:73b)
Some or all of the eigenvalues may be degenerate, but each eigenfunction has a
unique index i. Suppose further that the system is in state jÆ ji, one of the
eigenstates of A^. If we measure the physical observable A, we obtain the result
Æ j. What happens if we simultaneously measure the physical observable B? To
answer this question we need to calculate the expectation value hBi for this
system
hBi ˆ hÆ jjB^jÆ ji (3:74)
If we expand the state function jÆ ji in terms of the complete, orthonormal set
jâii
jÆ ji ˆ
X
i
cij âii
where ci are the expansion coefficients, and substitute the expansion into
equation (3.74), we obtain
3.11 Heisenberg uncertainty principle 99
hBi ˆ
X
i
X
k
c\ufffdk cihâkjB^jâii ˆ
X
i
X
k
c\ufffdk ciâiäki ˆ
X
i
jcij2 âi
where (3.73b) has been used. Thus, a measurement of B yields one of the many
values âi with a probability jcij2. There is no way to predict which of the values
âi will be obtained and, therefore, the observables A and B cannot both be
determined concurrently.
For a system in an arbitrary state Ø, neither of the physical observables A
and B can be precisely determined simultaneously if A^ and B^ do not commute.
Let \u2dcA and \u2dcB represent the width of the spread of values for A and B,
respectively. We define the variance (\u2dcA)2 by the relation
(\u2dcA)2 ˆ h(A^ÿ hAi)2i (3:75)
that is, as the expectation value of the square of the deviation of A from its
mean value. The positive square root \u2dcA is the standard deviation and is called
the uncertainty in A. Noting that hAi is a real number, we can obtain an
alternative expression for (\u2dcA)2 as follows:
(\u2dcA)2 ˆ h(A^ÿ hAi)2i ˆ hA^2 ÿ 2hAiA^‡ hAi2i
ˆ hA^2i ÿ 2hAihAi ‡ hAi2 ˆ hA^2i ÿ hAi2 (3:76)
Expressions analogous to equations (3.75) and (3.76) apply for (\u2dcB)2.
Since A^ and B^ do not commute, we define the operator C^ by the relation
[A^, B^] ˆ A^B^ÿ B^A^ ˆ iC^ (3:77)
The operator C^ is hermitian as discussed in Section 3.3, so that its expectation
value hCi is real. The commutator of A^ÿ hAi and B^ÿ hBi may be expanded
as follows
[A^ÿ hAi, B^ÿ hBi] ˆ (A^ÿ hAi)(B^ÿ hBi)ÿ (B^ÿ hBi)(A^ÿ hAi)
ˆ A^B^ÿ B^A^ ˆ iC^ (3:78)
where the cross terms cancel since hAi and hBi are numbers and commute with
the operators A^ and B^. We use equation (3.78) later in this section.
We now introduce the operator
A^ÿ hAi ‡ iº(B^ÿ hBi)
where º is a real constant, and let this operator act on the state function Ø
[A^ÿ hAi ‡ iº(B^ÿ hBi)]Ø
The scalar product of the resulting function with itself is, of course, always
positive, so that
h[A^ÿ hAi ‡ iº(B^ÿ hBi)]Øj[A^ÿ hAi ‡ iº(B^ÿ hBi)]Øi > 0 (3:79)
Expansion of this expression gives
100 General principles of quantum theory
h(A^ÿ hAi)Øj(A^ÿ hAi)Øi ‡ º2h(B^ÿ hBi)Øj(B^ÿ hBi)Øi
‡ iºh(A^ÿ hAi)Øj(B^ÿ hBi)Øi ÿ iºh(B^ÿ hBi)Øj(A^ÿ hAi)Øi > 0
or, since A^ and B^ are hermitian
hØj(A^ÿ hAi)2jØi ‡ º2hØj(B^ÿ hBi)2jØi
‡ iºhØj[A^ÿ hAi, B^ÿ hBi]jØi > 0
Applying equations (3.75) and (3.78), we have
(\u2dcA)2 ‡ º2(\u2dcB)2 ÿ ºhCi > 0
If we complete the square of the terms involving º, we obtain
(\u2dcA)2 ‡ (\u2dcB)2 ºÿ hCi
2(\u2dcB)2
\ufffd \ufffd2
ÿ hCi
2
4(\u2dcB)2
> 0
Since º is arbitrary, we select its value so as to eliminate the second term
º ˆ hCi
2(\u2dcB)2
(3:80)
thereby giving
(\u2dcA)2(\u2dcB)2 > 1
4
hCi2
or, upon taking the positive square root,
\u2dcA\u2dcB > 1
2
jhCij
Substituting equation (3.77) into this result yields
\u2dcA\u2dcB > 1
2
jh[A^, B^]ij (3:81)
This general expression relates the uncertainties in the simultaneous measure-
ments of A and B to the commutator of the corresponding operators A^ and B^
and is a general statement of the Heisenberg uncertainty principle.
Position\u2013momentum uncertainty principle
We now consider the special case for which A is the variable x (A^ ˆ x) and B
is the momentum px (B^ ˆ ÿi" d=dx). The commutator [A^, B^] may be evaluated
by letting it operate on Ø
[A^, B^]Ø ˆ ÿi" x dØ
dx
ÿ dxØ
dx
\ufffd \ufffd
ˆ i"Ø
so that jh[A^, B^]ij ˆ " and equation (3.81) gives
\u2dcx\u2dcpx >
"
2
(3:82)
The Heisenberg position\u2013momentum uncertainty principle (3.82) agrees
with equation (2.26), which was derived by a different, but mathematically
3.11 Heisenberg uncertainty principle 101
equivalent procedure. The relation (3.82) is consistent with (1.44), which is
based on the Fourier transform properties of wave packets. The difference
between the right-hand sides of (1.44) and (3.82) is due to the precise definition
(3.75) of the uncertainties in equation (3.82).
Similar applications of equation (3.81) using the position\u2013momentum pairs
y, p^y and z, p^z yield
\u2dcy\u2dcpy >
"
2
, \u2dcz\u2dcpz >
"
2
Since x commutes with the operators p^y and p^z, y commutes with p^x and p^z,
and z commutes with p^x and p^y, the relation (3.81) gives
\u2dcqi\u2dcpj ˆ 0, i 6ˆ j
where q1 ˆ x, q2 ˆ y, q3 ˆ z, p1 ˆ px, p2 ˆ py, p3 ˆ pz. Thus, the position
coordinate qi and the momentum component pj for i 6ˆ j may be precisely
determined simultaneously.
Minimum uncertainty wave packet
The minimum value of the product \u2dcA\u2dcB occurs for a particular state Ø for
which the relation (3.81) becomes an equality, i.e., when
\u2dcA\u2dcB ˆ 1
2
jh[A^, B^]ij (3:83)
According to equation (3.79), this equality applies when
[A^ÿ hAi ‡ iº(B^ÿ hBi)]Ø ˆ 0 (3:84)
where º is given by (3.80). For the position\u2013momentum example where A^ ˆ x
and B^ ˆ ÿi" d=dx, equation (3.84) takes the form
ÿi" d
dx
ÿ hpxi
\ufffd \ufffd
Ø ˆ i
º
(xÿ hxi)Ø
for which the solution is
Ø ˆ ceÿ(xÿhxi)2=2º"eih pxix=" (3:85)
where c is a constant of integration and may be used to normalize Ø. The real
constant º may be shown from equation (3.80) to be
º ˆ "
2(\u2dcpx)2
ˆ 2(\u2dcx)
2
"
where the relation \u2dcx\u2dcpx ˆ "=2 has been used, and is observed to be positive.
Thus, the state function Ø in equation (3.85) for a particle with minimum
position\u2013momentum uncertainty is a wave packet in the form of a plane wave
exp[ihpxix="] with wave number k0 ˆ hpxi=" multiplied by a gaussian
modulating function centered at hxi. Wave packets are discussed in Section
102 General principles of quantum theory
1.2. Only the spatial dependence of Ø has been derived in equation (3.85). The
state function Ø may also depend on the time through the possible time
dependence of the parameters c, º, hxi, and hpxi.
Energy\u2013time uncertainty principle
We now wish to derive the energy\u2013time uncertainty principle, which is
discussed in Section 1.5 and expressed in equation (1.45). We show in Section
1.5 that for a wave packet associated with a free particle moving in the x-
direction the product \u2dcE\u2dct is equal to the product \u2dcx\u2dcpx if \u2dcE and \u2dct are
defined appropriately. However,