Measurement in quantum mechanics problem

[jplcarpio]

Hi all,

In our assignment, we were given this question:

[PLAIN]http://img685.imageshack.us/img685/4854/prob213.png

I know that for (a), the answer is the first energy eigenstate since the measured energy corresponds to it. I'm not sure about the situation in (b), though.



Does "leaving the system alone, allowing it to evolve in the harmonic oscillator potential" mean that it returns to its original state (as given in the equation in the problem?

Or does it mean that a new state is created, following those in the harmonic oscillator potential, but with only the first energy eigenstate remaining?

I vaguely know that the act of measuring itself causes the wavefunction to collapse to a certain measurement and state, but what does the act of leaving it alone do?



I've tried to search through our reference, Introduction to Quantum Mechanics by David Griffiths, and through the Internet but so far I haven't seen material that might help me understand.

Thank you! :)

 

[vela]

Measurement collapses the wave function. What equation tells you how the wave function evolves in between measurements?

 

[jplcarpio]

I guess that would be the equation of the initial state, as given in the problem, since it's in the harmonic oscillator potential into which the system was allowed to evolve back. Would the initial state always hold in between measurements if it is left alone?

I've read that successive measurements would result in the same energy and energy eigenstate as the first measurement. Is that the only case when the measurement causes the wave function to change into specific eigenstates?

Thank you very much :)

 

[vela]

No, that's not correct. Try reading up on the Schrodinger equation.

 

[paranormal]

Hi there,

Measurement in quantum mechanics can be a tricky concept to understand. In this particular problem, it seems like the system is initially in the first energy eigenstate, and the question is asking what happens if we leave it alone to evolve in the harmonic oscillator potential.

In quantum mechanics, the wavefunction describes the state of a system. When we measure the energy of the system, we are essentially collapsing the wavefunction to a particular state (in this case, the first energy eigenstate). This is known as the "measurement problem" in quantum mechanics.

When we leave the system alone to evolve in the harmonic oscillator potential, the wavefunction will continue to evolve according to the Schrödinger equation. This means that the system will not necessarily return to its original state, but rather it will evolve to a new state that is a superposition of the energy eigenstates (including the first energy eigenstate). This is known as "quantum evolution".

So, in short, leaving the system alone does not necessarily mean that it will return to its original state, but rather it will evolve to a new state that is a superposition of the energy eigenstates. I hope this helps clarify the concept of measurement and evolution in quantum mechanics.