What is the expectation value for p in the given quantum mechanics problem?

[kde2520]

Homework Statement

First off, this is my first time posting here so please excuse any editing mistakes or guidelines I may have overlooked.

This is problem 1.17(c) from Griffiths, Introduction to Quantum Mechanics 2nd edition. It reads: $$\Psi$$(x, 0) = A(a^2 - x^2), -a$$\leq$$x$$\leq$$a. $$\Psi$$(x, 0) = 0, otherwise. What is the expectation value for p? (Note that you cannot get it from p = md<x>/dt. Why not?)

Homework Equations

So far we've derived the expression <p>=$$\int$$$$\Psi$$*($$\frac{h}{i}$$$$\frac{d}{dx}$$)$$\Psi$$dx

The Attempt at a Solution

I found the expectation value for position to be <x>=0. Also, t=0. These seem to explain why I can't get <p> from md<x>/dt. But since the function is not complex I can't see how to interpret the above expression for <p>. The operator acts on the real part, but there is no imaginary part to deal with. Any clues on how to interpret this?

 

[Domnu]

Are you talking about the $$\hbar / i$$ in the integration? You can pull this out of the integral, as this is just a constant. It's okay that there's no imaginary part... what would $$\psi^*$$ be in this case?

 

[kde2520]

Thanks for the reply. It wasn't the [tex]\hbar / i[tex]giving me trouble, but the idea of [tex]\psi^*$$being the conjugate of something with only real parts. Might sound silly I guess... Then I was having trouble seeing how we actually arived at the partial derivative with respect to x; but I see now that it was borrowed directly from Shrodinger's equation.<br /> <br /> Thanks again.$$[/tex][/tex][/tex]

 

[Domnu]

Any time =] But yes, the complex conjugate of a real function is just the function itself.

 

[lycogen]

eem
what about <p^2>?

i need to write d^2/dx^2? do I?

 

[Domnu]

You could do that, or you could also consider the Hamiltonian... what is the Hamiltonian in terms of <p^2>?

 

[lycogen]

i am doing the <p^2> for harmonic oscillator (ground state)
when I use second derivative I end up with part which contains x after integration :(

 

[Domnu]

okay if you're doing it for the harmonic oscillator, a *very* useful thing to do is to express all observables in terms of the annihilation/creation operators: a+ and a- in the book. go do that and see what happens.