708 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics
Solution
We use the fact thatψ 0 andψ 1 are normalized and orthogonal to each other.〈E〉1
2∫∞−∞(ψ∗ 0 +ψ∗ 1 )Ĥ(ψ 0 +ψ 1 )dx1
2∫∞−∞(ψ∗ 0 +ψ∗ 1 )(E 0 ψ 0 +E 1 ψ 1 )dx
1
2[
E 0∫∞−∞ψ∗ 0 ψ 0 dx+E 1∫∞−∞ψ∗ 0 ψ 1 dx+E 0∫∞−∞ψ∗ 1 ψ 0 dx+E 1∫∞−∞ψ∗ 1 ψ 1 dx]
1
2
(E 0 + 0 + 0 +E 1 )
1
2
(E 0 +E 1 )
1
2(
1
2
hv+
3
2
hv)
hv〈E^2 〉
1
2∫∞−∞(ψ∗ 0 +ψ∗ 1 )Ĥ^2 (ψ 0 +ψ 1 )dx
1
2∫∞−∞(ψ∗ 0 +ψ∗ 1 )(E^20 ψ 0 +E^21 ψ 1 )When we multiply out the integrand, there are four terms. The terms that contain bothψ 0
andψ 1 vanish after integration because they are orthogonal. The integral ofψ^20 orψ^21 equals
unity by normalization. We now have〈E^2 〉
1
2
(E^20 +E 12 )
5
4
(hv)^2σE(
〈E^2 〉−〈E〉^2) 1 / 2
[
5
4
(hv)^2 −(hv)^2] 1 / 2
hv
2This example shows that the statistical case applies if the wave function just prior to
the measurement is not an eigenfunction of the operator corresponding to the variable
being measured.
If the wave function at a certain time is given by a linear combination such as that in
Example 16.19, the time-dependent wave function is determined by this function and
the time-dependent Schrödinger equation.EXAMPLE16.20
If the wave function att0 is that of the preceding example, we know from Eq. (15.3-20)
that the wave function at timetisΨ(q,t)√
1
2(
ψ 0 (q)e−iE^0 t/h ̄+ψ 1 (q)e−iE^1 t/h ̄)a.Show that〈E〉is time-independent.
b.Show thatσEis time-independent.
Solution
a.〈E〉
1
2∫
(ψ∗ 0 (q)eiE^0 t/h ̄+ψ 1 ∗(q)eiE^1 t/h ̄)Ĥ(ψ 0 (q)e−iE^0 t/ ̄h+ψ 1 (q)e−iE^1 t/h ̄)dq
1
2∫
(ψ∗ 0 (q)eiE^0 t/h ̄+ψ 1 ∗(q)eiE^1 t/h ̄)(E 0 ψ 0 (q)e−iE^0 t/ ̄h+E 1 ψ 1 (q)e−iE^1 t/h ̄)dq