130_notes.dvi

As before, finite terms in the right hand integral go to zero asǫ→0, but now the delta function
gives a fixed contribution to the integral.

dψ

dx

∣

∣

∣

∣

+ǫ

−

dψ

dx

∣

∣

∣

∣

−ǫ

=

2 m

̄h^2

V 0 ψ(0)

There is adiscontinuity in the derivative of the wave functionproportional to the wave
function at that point (and to the strength of the delta function potential).

8.8 Examples

8.8.1 Hermitian Conjugate of a Constant Operator

If we have the operatorO=a+ibwhereaandbare real, what is its Hermitian conjugate? By the
definition of the Hermitian conjugate

〈φ|Oψ〉=〈O†φ|ψ〉.

It is easy to see from the integral that

〈(a−ib)φ|ψ〉=〈φ|(a+ib)ψ〉= (a+ib)〈φ|ψ〉

So the Hermitian conjugate of a constant operator is its complex conjugate.

8.8.2 Hermitian Conjugate of∂x∂

We wish to compute the Hermitian conjugate of the operator∂x∂. We will use the integral to derive
the result.
〈
φ

∣

∣

∣

∣

∂

∂x

ψ

〉

=

∫∞

−∞

φ∗(x)

∂ψ(x)

∂x

dx

We can integrate this by parts, differentiating theφand integrating to getψ.

〈

φ

∣

∣

∣

∣

∂

∂x

ψ

〉

= [φ∗(x)ψ(x)]∞−∞−

∫∞

−∞

∂φ∗(x)

∂x

ψ(x)dx=

〈

−∂

∂x

φ

∣

∣

∣

∣ψ

〉

So the Hermitian conjugate of∂x∂ is−∂x∂.

Note that the Hermitian conjugate of the momentum operator is−h ̄i−∂x∂ which is the same as the
original operator. So the momentum operator is Hermitian.

8.9 Sample Test Problems

1. A particle is confined to a box of lengthLin one dimension. It is initially in the ground state.
   Suddenly, one wall of the box is moved outward making a new box of length 3L. What is the
   probability that the particle is in the ground state of the new box? You may find it useful to