quantum mechanics questions

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A particle of mass is in the state



Where

a) Find .


I tried using the normalising condition but I keep getting





Am I missing something, or is the answer really this messy?


b) "Find "

Well,

how do you do that?!


Thanks!

[QuantumJG]

Give me until the end of march and I'll probably know how to do this. This maths seems awesome!

[mark_alec]

Look up the integral to a Gaussian.

[humph]

Look up the integral to a Gaussian.

+1.

You're right so far for the second, then make the substitution and use integration by parts twice.

I seem to remember helping my friends with questions like this two years ago...

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Are you sure the integral is elementary?



I tried integration by parts on it and also tried plugging it into my calculator but it won't work.



Hold on a sec... how do you do substitution with infinities?



,

,

Is this right?

[humph]

Hmmm, you're right, actually. Mathematica says the solution isn't elementary...

Also, the way to deal with the integral with that bit is to notice that it's even, so split it up around the origin into two integrals which are equal, then make the substitution.

EDIT: Actually this definite integral is still determinable, according to this. Indeed, we have

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Thanks humph!

(these problems are so mean )

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The Hamiltonian-Jacobi equation



Can be re-expressed as by taking

Schrodinger says that if we think of the level curves of W, and assign an arbitrary curve the value , that we can take a normal to that paticular level curve (spanning ) to be":



(In other words, )

Where does this come from? How do we know the normal differential has this value?

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Can someone please help me solve:



It comes from separating the schrodinger equation in cylindrical coordinates.

[mark_alec]

From memory, let u(r) = 1/R and convert your DE to one in terms of du/dr. It will work out to be much nicer.

[humph]

You could always just multiply through by , so it becomes a 2nd order linear ODE. Then try a power series solution perhaps, or any other technique to solve 2nd order linear ODEs (I'm not sure what you're up to in Methods 1...).

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From memory, let u(r) = 1/R and convert your DE to one in terms of du/dr. It will work out to be much nicer.

thanks mark, I'll have a go at that

You could always just multiply through by , so it becomes a 2nd order linear ODE. Then try a power series solution perhaps, or any other technique to solve 2nd order linear ODEs (I'm not sure what you're up to in Methods 1...).

yeah I thought that might work but if possible I want to try to get an analytic solution first

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Too busy at the moment for stuff that's outside the course... I'll try the DE at some later... indefinite date.

Speaking of indefiniteness, to find the momentum distribution of a particle in an infinite square well, do you:

a) Decompose into and by noting that the momentum operator operating on each of the exponentials gives back an eigenvalue for momentum, conclude that the momentum probability distribution is given by two spikes at ?

OR

b) Use the Fourier Transform to find the momentum distribution which has two peaks but smooth peaks, rather than the discrete spikes given in the first method?

This is from an assignment I just handed in. I went with the second method since I trust the Fourier Transform and it seems to make more sense to have continuous position/momentum fourier pairs. but a few of my friends argued using the first method that the momentum has discrete values.

Which one is correct? thx

[mark_alec]

http://mathworld.wolfram.com/FourierTransformSine.html

If you do the Fourier transform, the answer looks strangely like what you'd get if you did it the first way. Indeed, you expect the particle is travelling in one direction, or the other, at a speed dependent upon the energy.

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But why is it the dirac delta function? When I integrated the exponentials I got cosines and signs.