What Are the Probable Values of Energy and Momentum for a Free Particle at t=0?

[nikolafmf]

Homework Statement

At time t=0 free particle is found in state psi=const*sin(3x)*exp[i(5y+z)]. What values for energy and for momentum we can get if we measure them at t=0 and with what probability?

Homework Equations

Well, we know that eigenvalues of energy and momentum operator for free particle are (hbar*k)^2/(2m) and hbar*k respectively.

The Attempt at a Solution

So, does measured momentum equal hbar*(0, 5, 1) and energy 13(hbar)^2/m? Is probability 100%? Book hints that we can get different values for momentum, not just one I stated before. How is it possible?

 

[BruceW]

You've got the y and z momentum correct. But You've not got the correct x momentum. In the wavefunction, the bit that depends on x is: sin(3x) So you've got to think of how to express this in terms of eigenstates.

 

[nikolafmf]

With the use of Fourier transform? But if my limits of integration are infinity, I get diverging integrals. What limits should I use?

 

[BruceW]

you don't need to do anything complicated. I'm sure you must have learned about how to write a sine function as a sum of two complex exponentials... (that's the hint)