more subtle and a full treatment will not be given here. Here is what can be proved. For any vector Ψ L2(R) we can find coe cients an such that

n=0

Since this is an infinite sum we must say in what sense this sum converges. If we define the partial sums

n

X

Ψn = ak ψk , k=0

then we say Ψn → Ψ as n → ∞ if

lim k Ψ − Ψn k= 0.

n→∞

(Observe that Ψn, being a sum of a finite number of terms is well-defined.) Recall that the norm k · k in L2(R) is

Z

k Ψ − Ψn k2= |Ψ(x) − Ψn(x)|2 dx.

R

It is in this sense the series converges. Since ψn form an orthonormal sequence, the coe - cients an are given simply by

an = (ψn, Ψ) .

Observe that since ψn form an orthonormal basis, the vector Ψ in (7.22) satisfies

∞

k Ψ k2= X |an|2 < ∞.

n=0

7.3Some properties of the harmonic oscillator

In quantum mechanics if O is an observable (mathematically, a self-adjoint operator on the Hilbert space H), then the average (or expected) value of O in the state Ψ is

hOi = (OΨ, Ψ) .

For the quantum oscillator, the average position in the eigenstate ψn is

Z

hxi = (xψn, ψn) = xψn(x)2 dx = 0.

R

(The integral is zero since ψn2 is an even function of x so that xψn(x)2 is an odd function.) The average of the square of the position in the eigenstate ψn is

x2 = x2ψn, ψn .

108 CHAPTER 7. QUANTUM HARMONIC OSCILLATOR

This inner product (integral) can be evaluated by first using (7.20) twice to write x2ψn as a linear combination of the ψk’s:

The inner product can now be calculated using the orthonormality of the wave functions to find18

x2

A very similar calculation with pˆ = −i dxd (but this time using (7.21)) gives19

hpˆi = 0,

pˆ2 = (n + 12 ) .

If we define

q

x= hx2i − hxi2 ,

q

p= hpˆ2i − hpˆi2 ,

then (in physical units) we have in state ψn

This is the Heisenberg Uncertainty Principle for the harmonic oscillator. The inequality part of the statement can be shown to be valid under very general conditions.

7.3.1Averages hxˆ(t)i and hpˆ(t)i

Let Ψ be any state of the system

∞

X

Ψ(x) = anψn(x)

n=0

18This is the dimensionless result. Putting back in the dimensions, the average is

~

(n + 1/2)

mω0

.

19Again these are the dimensionless results. Putting back the units the second average is

= (n + 1 ) mω0~. 2

are convergent sums. (Here, as throughout, the ψn are the harmonic oscillator wave functions.) The time evolution of Ψ is then given by

n≥0

which follows from the above discussion of separation of variables.

In the state Ψ(x, t) we are interested in computing the average values of xˆ and pˆ. For notationally convenience let’s define

xAVG (t) = hxˆi = (ˆxΨ(x, t), Ψ(x, t))

and

pAVG(t) = hpˆi = (pˆΨ(x, t), Ψ(x, t)) .

Let

x0 := xAVG (0) = (ˆxΨ(x, 0), Ψ(x, 0))

and

p0 := pAVG (0) = (pˆΨ(x, 0), Ψ(x, 0)) .

We first calculate x0 and p0.

=A + B

where we use the orthonormality of the functions ψn. Similarly,

p0 = −iA + iB.

We now calculate xAVG (t) Now the state is (7.23). Proceeding as in the t = 0 case we see

X

xAVG (t) = ana¯me−i(EN −EM )t/~ (xψn, ψm) .

m,n≥0

The calculation of the inner products (xψn, ψm) was done in the t = 0 case. Noting that