Time independant schrodinger equation

[leoflindall]

Homework Statement

Write down the time independant schrodinger equation in the momentum representation for a particle with mass m when the potential is given by V (x) = 1/2 [tex]\gamma[/tex] x².

A possible soloution of this schrodinger equation is of the form

[tex]\Psi[/tex] (p) = Ae^{-Bp2 / 2}

Determine B and the corresponding energy eigenvalue.

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Homework Equations

The Attempt at a Solution

I think the schrodinger equation is

[tex]\frac{p2}{2m}[/tex] [tex]\Psi[/tex] (r) - 1/2 [tex]\gamma[/tex] [tex]\hbar[/tex]² d2/dx2 [tex]\Psi[/tex] (r) = E [tex]\Psi[/tex] (r)

Where i have used the momentum representation p = p . x = -ih d/dx.

Is this correct?

Secondly how do approach determing B, and find the energy eigenvalue?

Do i need to solve the equation directly, possibly by using separation of vartiables?

Many thanks for any advice or help.

 

[SpiffyKavu]

You mostly wrote down the correct Schrodinger equation in the momentum representation. The wavefunction is a function of momentum, not position.

[tex]
\frac{p^2}{2m}\varphi(p) - \frac{\hbar^2\gamma}{2}\left(\frac{d}{dx}\right)^2 \varphi(p) = E\varphi(p)
[/tex]

You are given a wavefunction, and you want to determine what value of B gives you an eigenfunction of the above equation. The most straightforward way of doing this is just to do the math. Put your wavefunction into the above equation. The right side (E) needs to be a constant, or else your wavefunction is not an eigenfunction of the equation.

When in doubt, do the math. You just need to keep in mind what you are looking for (eigenfunction), and its corresponding definition.