What is the Mean Number of Oscillatory Quanta After a Hamiltonian Change?

[upender singh]

A quantum mechanical oscillator with the Hamiltonian
H1=p^2/2m +(m(w1)^2 x^2)/2

is initially prepared in its ground state (zero number of oscillatory quanta). Then the
Hamiltonian changes abruptly (almost instantly):
H1→H2=p^2/2m +(m(w2)^2 x^2)/2
What is the mean number of oscillatory quanta upon the transformation?My first question is what does oscillatory quanta exactly means?

Attempt: Theory of quantum harmonic oscillator, the eigenstate formulas, the energy formulas. The only thing that is zero in ground state is n=0, so does it mean oscillatory quanta implies n quantum number.

 

[PeroK]

You should have posted this in the homework forum, as you'll get a better response.

Let me explain the question at least. The oscillator is in a known state (ground state for first Hamiltonian). The Hamiltonian changes, which means that the energy eigenstates change. Now you have effectively an initial value problem. You know the initial state/wave function, and this you need to express as a linear combination of your new eigenstates.

I suspect the mean oscillatory quanta means the expected value of $n$ in your new system. Where $n$ represents the energy levels in your new system.

 

[upender singh]

Hi Perok,
Sorry for posting it at wrong place.
Do you mean that the my initial state is the ground state of the old Hamiltonian. Now since the Hamiltonian has changed, I need to express it(ground state from old Hamiltonian) as a combination of the eigenstates of new Hamiltonian?

 

[PeroK]

Hi Perok,
Sorry for posting it at wrong place.
Do you mean that the my initial state is the ground state of the old Hamiltonian. Now since the Hamiltonian has changed, I need to express it(ground state from old Hamiltonian) as a combination of the eigenstates of new Hamiltonian?

Yes, that's what you have here.