Particle bound by quadratic potential

[kasse]

Homework Statement

A particle that can move in one dimension and that is in a stationary state, is bound by a potential V(x) = (1/2)kx^2. The wave function is \Psi(x,t) = \psi(x)exp(-iEt/\hbar)

We look at a state in which \psi(x) = Aexp(-x^2/2a^2a^2), where a is a constant and A is the normalisation constant. Determine a so that \psi(x) is an energy state. What is the energ of the particle?

The Attempt at a Solution

I don't really know what to do here, but setting up the wave equation tends to be a good start:

H\psi(x) = E\psi(x), where H = -(\hbar^2/2m)(d^2/dx^2) + kx^2/2

How can I determine a so that \psi(x) is an energy state?

 

[borgwal]

Just calculate the left-hand and right-hand sides, and set them equal.

 

[kasse]

That the wave function is an energy state simply means that it's described by the Schrödinger equation?

 

[borgwal]

Yes, it's a solution to the *time-independent* SE, to be precise.

 

[kasse]

OK, but I still don't see how to find a when E is unknown. My expression after differentiating twice is:

(\hbar^2/2ma^2) (A - x^2/a^2) + kx^2/2 = E

 

[borgwal]

First, you didn't calculate it right.

Second, the correct equation you'll get will have to be true for every x.

 

[kasse]

Now I ended up with

(\hbar^2/2ma^2)(1 - x^2/a^2) + kx^2/2 = E

What I should do here is to eliminate x so that the equation holds for all x-es? In that case, a must be (\hbar^2/km)^0.25

 

[borgwal]

Your equation is now of the form (if you move E to the left-hand side)

C+Dx^2=0,

where C and D are constants (independent of x).
If this is to be true for all x, then both C and D must be zero.

 

[kasse]

I got it right now, didn't I?

My next problem is to calculate the expectation value. That's not hard, I just wonder why the integral is written

\intdx \psi*(x)x\psi(x)

and not

\int \psi*(x)x\psi(x) dx

 

[f95toli]

In this case it doesn't matter where you put the "dx", it is the same thing.

(note that there are cases where it DOES matter; e.g. if dx is a vector)