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Q.M. harmonic oscillator spring constant goes to zero at t=0

  • #1

Homework Statement



A one-dimensional harmonic oscillator is in the ground state. At t=0, the spring is cut. Find the wave-function with respect to space and time (ψ(x,t)).

Note: At t=0 the spring constant (k) is reduced to zero.


So, my question is mostly conceptual. Since the spring constant goes to zero at t=0 is it safe to assume that the problem can now be considered as a free particle problem since the potential goes to zero when 'k' goes to zero?

If this assumption is correct I should be able to solve the time independent Schrodinger equation, ψ(x,0), then multiply in the time part and solve for the constant.

I do not see how being a harmonic oscillator will affect my answer since once you cut the spring it is no longer a harmonic oscillator.

Is there something I am missing conceptually?
 

Answers and Replies

  • #2
TSny
Homework Helper
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Hello FarticleFysics and welcome to Physics Forums!

So, my question is mostly conceptual. Since the spring constant goes to zero at t=0 is it safe to assume that the problem can now be considered as a free particle problem since the potential goes to zero when 'k' goes to zero?]
Yes, the particle instantaneously becomes free at t = 0. But the wavefunction does not undergo any instantaneous change at t = 0.

##\psi(x, 0^+)## = ##\psi(x, 0^-)##

If this assumption is correct I should be able to solve the time independent Schrodinger equation, ψ(x,0), then multiply in the time part and solve for the constant.
The wavefunction at time t = 0 will not be a solution of the time independent Schrodinger equation for a free particle. But the wavefunction at t = 0 may be expanded as a superposition of solutions of the time independent, free-particle Schrodinger equation. You can then put in a time dependent factor for each member of the superposition.

I do not see how being a harmonic oscillator will affect my answer since once you cut the spring it is no longer a harmonic oscillator.
The harmonic oscillator potential can be thought of as "preparing" the quantum state of the particle at time t = 0.
 
Last edited:

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