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

• upender singh
In summary: You are essentially looking for the expectation value of the number operator in the new system, using the initial state as your wave function.In summary, the problem involves a quantum mechanical oscillator initially in ground state and then undergoing a sudden change in Hamiltonian, resulting in a new system with different energy eigenstates. The mean number of oscillatory quanta in the new system is determined by expressing the initial state as a linear combination of the new eigenstates and finding the expectation value of the number operator.
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.

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.

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?

upender singh said:
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.

## 1. What is the meaning of "mean number of oscilatory quanta"?

The mean number of oscilatory quanta refers to the average number of quantized energy units or particles that are involved in the oscillatory motion of a system, such as a vibrating string or an atom.

## 2. How is the mean number of oscilatory quanta calculated?

The mean number of oscilatory quanta is calculated by dividing the total energy of the system by the energy of a single quantum. This can also be expressed as the ratio of the frequency of the oscillation to the energy of a single quantum.

## 3. What is the significance of the mean number of oscilatory quanta?

The mean number of oscilatory quanta is a useful measure in understanding the behavior of a system in terms of quantized energy levels. It can also provide insights into the stability and energy distribution of the system.

## 4. Can the mean number of oscilatory quanta change?

Yes, the mean number of oscilatory quanta can change depending on the energy and frequency of the system. For example, increasing the energy of the system would result in a higher mean number of quanta, while decreasing the frequency would result in a lower mean number of quanta.

## 5. How is the mean number of oscilatory quanta related to the uncertainty principle?

The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. The mean number of oscilatory quanta is related to this principle as it represents the average energy of a system, which is a measure of its momentum. Therefore, the more energy a system has, the more uncertain its position becomes.

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