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

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Homework Help Overview

The discussion revolves around a free particle described by a wavefunction at time t=0, specifically psi=const*sin(3x)*exp[i(5y+z)]. Participants are exploring the possible values of energy and momentum that can be measured, along with their associated probabilities.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between the wavefunction and the eigenvalues of energy and momentum operators. There is an attempt to determine specific momentum values and energy based on the given wavefunction, while questioning the implications of the book's hint about multiple momentum values.

Discussion Status

Some participants have confirmed the momentum values for the y and z components but have raised concerns about the x component. There is a suggestion to use Fourier transforms to analyze the wavefunction, although one participant expresses difficulty with diverging integrals. Another participant hints at a simpler method involving the representation of sine functions.

Contextual Notes

Participants are grappling with the implications of the wavefunction's form and the potential for multiple momentum values, as well as the mathematical techniques required to analyze it. There is an indication of homework constraints that may limit the methods available for solving the problem.

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

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