Harmonic oscillator with ladder operators - proof using the Sum Rule

In summary, the conversation discusses verifying the proof of the sum rule for the one-dimensional harmonic oscillator using ladder operators and expressing the momentum operator in terms of the ladder operators. The first step is to calculate the left side of the sum rule using the ladder operators, and the hint suggests using the Hamiltonian operator and the energy eigenvalue equation. The conversation also touches on using commutation relations and generalizing the proof for a general operator.
  • #1
chocopanda
15
1
Homework Statement
Verify the proof of the sum rule for the one-dimensional harmonic oscillator:
$$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$
Relevant Equations
The exercise explicitly says to use laddle operators and to express $p$ with
$$b=\sqrt{\frac {mw}{2 \hbar}}-\frac {ip}{\sqrt{2 \hbar mw}} $$
$$b^\dagger =\sqrt{\frac {mw}{2 \hbar}}+\frac {ip}{\sqrt{2 \hbar mw}} $$
I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator:
$$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$
The exercise explicitly says to use laddle operators and to express $p$ with
$$b=\sqrt{\frac {mw}{2 \hbar}}-\frac {ip}{\sqrt{2 \hbar mw}} $$
$$b^\dagger =\sqrt{\frac {mw}{2 \hbar}}+\frac {ip}{\sqrt{2 \hbar mw}} $$

For p I get $$p=i \sqrt{\frac{\hbar}{2mw}} (b-b^\dagger) $$

To solve the exercise, we need to calculate the left side. I'm still very much a novice and am not very sure how to use the ladder operators... To start, I at least tried to expand the bra-ket:
$$\sum_l^\infty (E_l-E_n)\ \langle l \ |p| \ n \rangle \langle n \ |p| \ l \rangle $$
and tried to insert the p I solved:
$$\sum_l^\infty (E_l-E_n)\ (-\frac{\hbar}{2mw}) \langle l \ |b-b^\dagger| \ n \rangle \langle n \ |b-b^\dagger| \ l \rangle $$
is this correct? If yes, how do I continue? The hint says to probably use $$H|n\rangle=\hbar(n+\frac 12)|n\rangle$$ and I know that $$H|n\rangle=E|n\rangle$$
 
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  • #2
chocopanda said:
To start, I at least tried to expand the bra-ket:
$$\sum_l^\infty (E_l-E_n)\ \langle l \ |p| \ n \rangle \langle n \ |p| \ l \rangle $$
I wouldn't do that right away. I would start by calculating ##\langle l \ |p| \ n \rangle## first, and then consider the absolute value squared.

chocopanda said:
and tried to insert the p I solved:
$$\sum_l^\infty (E_l-E_n)\ (-\frac{\hbar}{2mw}) \langle l \ |b-b^\dagger| \ n \rangle \langle n \ |b-b^\dagger| \ l \rangle $$
is this correct? If yes, how do I continue?
Calculate ##(b-b^\dagger) |n\rangle##.
 
  • #3
DrClaude said:
I wouldn't do that right away. I would start by calculating ##\langle l \ |p| \ n \rangle## first, and then consider the absolute value squared.Calculate ##(b-b^\dagger) |n\rangle##.

Hello DrClaude, thank you for replying. I tried to do what you suggested:

$$| \langle l|p|n \rangle |^2 = \langle n|p^2|n\rangle = \frac{h}{2mw} (2n+1) $$
That's my result. How would I continue?
 
  • #4
chocopanda said:
Hello DrClaude, thank you for replying. I tried to do what you suggested:

$$| \langle l|p|n \rangle |^2 = \langle n|p^2|n\rangle = \frac{h}{2mw} (2n+1) $$
That's my result. How would I continue?
That's not correct. How can the bra ##\langle l |## even become ##\langle n |##?

As I said, forget the absolute value squared for now. Start by calculating ##(b-b^\dagger) |n\rangle## and then apply that to ## \langle l|p|n \rangle##.
 
  • #5
Are you forced to do it in that complicated way? It's simpler to use the commutation relations, ##[\hat{x},\hat{p}]=?##, ##[\hat{H},\hat{x}]=?## as well as to think about what's
$$\sum_{i} |\langle n|\hat{O}|i \rangle|^2=?$$
for a general operator ##\hat{O}##.
 
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1. What is a harmonic oscillator?

A harmonic oscillator is a system in which the restoring force is proportional to the displacement from the equilibrium position. It is a common model used in physics to describe the behavior of many physical systems, such as a mass on a spring or a pendulum.

2. What are ladder operators in the context of a harmonic oscillator?

Ladder operators are mathematical operators used to describe the energy levels of a quantum mechanical system, such as a harmonic oscillator. They allow us to calculate the energy of the system at different levels and to move between these energy levels.

3. How does the Sum Rule relate to the proof of the harmonic oscillator using ladder operators?

The Sum Rule is a mathematical principle that states that the sum of the probabilities of all possible outcomes of an event is equal to 1. In the context of the harmonic oscillator, the Sum Rule is used to show that the total energy of the system can be expressed as a sum of the energies at each energy level, which is a key step in the proof using ladder operators.

4. What is the significance of the proof of the harmonic oscillator using ladder operators?

The proof using ladder operators is significant because it provides a mathematical framework for understanding the energy levels and behavior of a harmonic oscillator. It also allows us to make predictions about the behavior of other quantum mechanical systems that can be described using ladder operators.

5. Is the proof of the harmonic oscillator with ladder operators applicable to all harmonic oscillators?

Yes, the proof using ladder operators is applicable to all harmonic oscillators, regardless of the specific physical system being modeled. This is because the fundamental principles and mathematical framework of the proof apply to all harmonic oscillators.

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