Potential and kinetic energies in Quantum Oscillator

In summary, potential energy is the energy stored in a system's position or configuration, while kinetic energy is the energy associated with its movement. In a Quantum Oscillator, potential energy is represented by the potential well and kinetic energy by the particle's motion within it. The potential energy changes as the particle moves due to the changing shape of the potential well. The potential and kinetic energies are interrelated through energy conservation, allowing the particle to constantly oscillate between them. The energy level of a Quantum Oscillator is determined by the quantum number n. Potential and kinetic energies cannot be measured separately in a Quantum Oscillator due to the uncertainty principle in quantum mechanics. They are measured together as the total energy of the system.
  • #1
alchemistoff
8
0

Homework Statement



Problem 9. Evaluate the matrix elements [tex]\langle n + \nu|x^2|n\rangle[/tex] and [tex]\langle n + \nu|p^2|n\rangle[/tex] in
the harmonic oscillator basis, for [tex] \nu = 1, 2, 3, 4[/tex] :
1. Using the closure property and the matrix elements.
2. Applying the operators [tex] x^22[/tex] and [tex] p^2[/tex] , expressed in terms of the [tex] a+, a[/tex] on the eigenstates.
3. Find the ratio [tex] \langle n + \nu|K|n\rangle/\langle n + \nu|V|n\rangle[/tex] [tex] (\nu = 0, \pm2)[/tex] between the kinetic
and the potential energy matrix elements. Justify the differences in sign
on quantum mechanical grounds.


Homework Equations



[tex]H=\frac{p^2}{2m}+\frac{m\omega ^2}{2}x^2[/tex]

The Attempt at a Solution



[tex] \frac{\langle n|K|n \rangle}{\langle n|V|n \rangle}=-\frac{\langle n\pm 2|K|n \rangle}{\langle n\pm 2|V|n \rangle}=1[/tex]
...but I cannot justify the difference in sign on quantum mechanical grounds!
 
Physics news on Phys.org
  • #2

Thank you for your post. Let me first clarify the problem statement for those who may not be familiar with the notation. The problem is asking you to evaluate the matrix elements of the operators x^2 and p^2 in the harmonic oscillator basis, for different values of the parameter ν. The second part of the problem asks you to find the ratio of the kinetic energy matrix element to the potential energy matrix element for certain values of ν.

To address your attempt at a solution, you have correctly evaluated the ratio of the kinetic and potential energy matrix elements for ν = 0 and ν = ±2. However, you are correct in saying that you cannot justify the difference in sign on quantum mechanical grounds. This is because the operators x^2 and p^2 have different signs in the Hamiltonian, as shown in the homework equation. This leads to a difference in sign in the matrix elements for these operators.

To better understand this, we can look at the operators x and p individually. The operator x represents the position of a particle, and the operator p represents its momentum. In classical mechanics, the kinetic energy is given by p^2/2m, where m is the mass of the particle. In quantum mechanics, the operator p^2 is replaced by -ħ^2(d^2/dx^2), which is why there is a difference in sign between the classical and quantum expressions for kinetic energy.

Similarly, the potential energy in classical mechanics is given by mω^2x^2/2, where ω is the frequency of the harmonic oscillator. In quantum mechanics, the operator x^2 is replaced by x^2, but the potential energy still retains its classical form. This is why there is no difference in sign between the classical and quantum expressions for potential energy.

In conclusion, the difference in sign between the kinetic and potential energy matrix elements is a result of the difference in sign between the operators x^2 and p^2 in the Hamiltonian. This difference in sign is due to the different representations of kinetic energy in classical and quantum mechanics.

I hope this explanation helps to clarify your understanding. If you have any further questions, please don't hesitate to ask.
 

1. What is the difference between potential and kinetic energy in a Quantum Oscillator?

Potential energy is the energy stored in the position or configuration of a system, while kinetic energy is the energy associated with the movement of the system. In a Quantum Oscillator, potential energy is represented by the potential well in which the particle is confined, while kinetic energy is represented by the particle's motion within the potential well.

2. How does the potential energy change as the particle moves in a Quantum Oscillator?

In a Quantum Oscillator, the potential energy changes as the particle moves because the potential well changes shape. As the particle moves towards the edges of the well, the potential energy increases, and as it moves towards the center, the potential energy decreases.

3. What is the relationship between potential and kinetic energy in a Quantum Oscillator?

The potential and kinetic energies in a Quantum Oscillator are interrelated through the principle of energy conservation. As the potential energy increases, the kinetic energy decreases and vice versa. This relationship allows the particle to constantly oscillate between potential and kinetic energies.

4. How is the energy level of a Quantum Oscillator determined?

The energy level of a Quantum Oscillator is determined by the quantum number n, which represents the number of energy levels the particle can occupy. The higher the value of n, the higher the energy level of the particle.

5. Can potential and kinetic energies in a Quantum Oscillator be measured separately?

No, potential and kinetic energies in a Quantum Oscillator cannot be measured separately. This is due to the uncertainty principle in quantum mechanics, which states that the more accurately we measure one aspect of a particle's motion, the less accurately we can measure another aspect. Therefore, both potential and kinetic energies are measured together as the total energy of the system.

Similar threads

Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
5
Views
806
  • Advanced Physics Homework Help
Replies
10
Views
460
  • Advanced Physics Homework Help
Replies
13
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
871
Replies
4
Views
1K
Replies
16
Views
397
  • Advanced Physics Homework Help
Replies
10
Views
1K
  • Advanced Physics Homework Help
Replies
15
Views
2K
Back
Top