Superposition Principle Question

[Ed Quanta]

So if we have a particle in a one dimensional box with walls at x=0 and x=a. Now suppose one of the walls is moved in a time short compared to the natural period 2pi/w1, where (h/2pi)w1=E. If the energy of the particle is measured soon after this expansion, what value of energy is most likely to be found. How does this energy compare to the particle's initial energy E1?

Help anyone? I am not sure how to go about solving this.

I usually use the equation Psi(x,0)= the integral of b(k)(Psik)dk

where Psik=(1/square root of (2pi))exp(ikx)


But since I am not told what Psi(x,0) is, there is no way for me to solve for
b(k).

Am I just interpreting this problem completely wrong?

 

[ZapperZ]

So if we have a particle in a one dimensional box with walls at x=0 and x=a. Now suppose one of the walls is moved in a time short compared to the natural period 2pi/w1, where (h/2pi)w1=E. If the energy of the particle is measured soon after this expansion, what value of energy is most likely to be found. How does this energy compare to the particle's initial energy E1?

Help anyone? I am not sure how to go about solving this.

I usually use the equation Psi(x,0)= the integral of b(k)(Psik)dk

where Psik=(1/square root of (2pi))exp(ikx)


But since I am not told what Psi(x,0) is, there is no way for me to solve for
b(k).

Am I just interpreting this problem completely wrong?

I think there's at least 2 different ways to look at this problem, which means it may be a bit vague. In any case, the link below may have the method that you need to solve the problem.

http://electron6.phys.utk.edu/phys594/archives/quantum/Postulates/postulates1.htm

This really should have gone into the Homework help section.

Zz.

 

[R0man]

The superposition principle states that the total wave function of a system is the sum of all individual wave functions present in the system. In this case, the particle in the one dimensional box can be described by a wave function, Psi(x), which is a superposition of all the possible energy states of the particle.

When one of the walls is moved, the potential energy of the particle changes, and hence its energy state also changes. However, the superposition principle still holds true, and the new wave function can be written as a superposition of the initial wave function and the wave function corresponding to the new potential energy.

Now, the question asks what value of energy is most likely to be found when the energy of the particle is measured soon after the expansion. This can be answered by looking at the amplitude of the new wave function at different energy states. The energy state with the highest amplitude will be the most likely energy state to be found upon measurement.

In this case, the most likely energy state will be the one closest to the initial energy E1. This is because the expansion of the wall is short compared to the natural period of the system, and hence the change in energy will be small. Therefore, the energy of the particle is most likely to be found to be E1.

This can also be seen by comparing the energy states of the initial wave function and the new wave function. Since the expansion is short and the change in energy is small, the energy states of the new wave function will be very similar to the energy states of the initial wave function. Hence, the energy of the particle is most likely to be found to be E1.

In summary, the energy of the particle is most likely to be found to be its initial energy E1 when measured soon after the expansion of one of the walls. This is because of the superposition principle and the fact that the expansion is short and the change in energy is small.