Subatomic Physics

198 Additive Conservation Laws
To evaluate the last expression, the complex conjugate Schrödinger equation is
needed:
−i� dψ∗
dt =(Hψ)∗ = ψ ∗ H. (7.2)
Here the reality of H has been used. With Eqs (7.1) and (7.2), (d/dt)〈F 〉 becomes
�
d i
〈F 〉 =
dt �
d 3 xψ ∗ (HF − FH)ψ. (7.3)
The term HF − FH is called the commutator of H and F and it is denoted by
brackets:
HF − FH ≡ [H, F]. (7.4)
Equation (7.3) shows that 〈F 〉 is conserved (i.e., is a constant of the motion) if the
commutator of H and F vanishes:
[H, F] =0→ d
〈F 〉 =0. (7.5)
dt
If H and F commute, the eigenfunctions of H can be chosen so that they are also
eigenfunctions of F ,
Hψ = Eψ
Fψ = fψ.
(7.6)
Here, E is the energy eigenvalue and f the eigenvalue of the operator F in the state
ψ.
for instance, Chapter 8 of Merzbacher—will remove the problem. We only remark that an observable
is represented by a quantum mechanical operator F whose expectation value corresponds to
a measurement. The expectation value of F in the state ψa is defined as
�
〈F 〉 = d 3 xψ ∗ a (x)Fψa(x).
Since the expectation value of F can be measured, it must be real, and F therefore must be
Hermitian. If two states are considered, a quantity similar to 〈F 〉 can be formed by writing
�
Fba = d 3 xψ ∗ b (x)Fψa(x).
Fba is called the matrix element of F between states a and b. The expectation value of F in state
a is the diagonal element of Fba for b = a:
〈F 〉 = Faa.
The off-diagonal elements do not correspond directly to classical quantities. However, transitions
between states a and b are related to Fba (Merzbacher, Section 5.4).