Quantum mechanics: Quantum particle in a harmonic oscillator potential motion

In summary, the conversation discusses a problem involving a quantum particle moving in a harmonic oscillator potential and the corresponding eigenstates and wave functions. The main points of the conversation include finding the constant A, obtaining the wave function at a later time, calculating the probability density, and calculating the expectation value of the energy. The use of the time evolution operator and infinite geometric series is mentioned as a method for solving the problem.
  • #1
rapupaux
1
0
I'm sorry if the form of my post does not meet the general requirements(this is the first time i work with any kind of LaTeX) and I promise that my next posts will be more adequate. Right now I am in serious need of someone explaining me this problem, since on the 6th of June I'm supposed to present it to my QM professor for extra points in the exam given on the same day.

Any help will be much appreciated!
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A quantum particle is moving in a harmonic oscillator potential [tex]V(x)=\frac{m\omega^{2}x^{2}}{2}[/tex].The eigenstates are denoted by |n> while the wave functions are: [tex]\Psi_{n}(x)=<x|n>[/tex].

At t=0 the system is in the state:

[tex]|\Psi (t=0) > = A \sum_{n} (\frac{1}{\sqrt{2}})^{n}| n> [/tex]

1) Find the constant A
2) Obtain the expression for the wave function at a latter time: [tex]\Psi(x,t)\equiv<x|\Psi(t)>[/tex]
3) Calculate the probability density: [tex] | \Psi(x,t) |^{2} [/tex]
4) Calculate the expectation value of the energy.
 
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  • #2
for the 1) normalise psi and you get an infinite geometric series so look up infinite geometric series

for 2) use the time evolution operator [tex] e^{-iHt/ \hbar} [/tex] where H is the Hamiltonian which acting on eigenstates just replaces the operator with the eigenvalue and again you get an infinite geometric series

do you know how to do the rest?
 

1. What is quantum mechanics?

Quantum mechanics is a branch of physics that deals with the behavior and interactions of particles at the atomic and subatomic level. It explains how particles, such as electrons and photons, behave and interact with each other in ways that are not explained by classical physics.

2. What is a harmonic oscillator potential?

A harmonic oscillator potential is a mathematical model that describes the potential energy of a particle in a system that exhibits simple harmonic motion. It is a type of potential energy well that has a parabolic shape and represents the energy states available to a particle within a certain system.

3. How does a quantum particle behave in a harmonic oscillator potential?

A quantum particle in a harmonic oscillator potential will exhibit quantized energy levels, meaning that the particle can only have certain discrete energy values. The particle will also exhibit wave-like behavior, with a probability distribution of its position and momentum rather than a definite position and momentum.

4. What is the significance of the ground state in a quantum harmonic oscillator?

The ground state in a quantum harmonic oscillator is the lowest energy state that a particle can have within the potential. It has the lowest energy and the particle will have a high probability of being found in this state. The ground state is also important because it forms the basis for understanding higher energy states and transitions between them.

5. How is the quantum harmonic oscillator used in practical applications?

The quantum harmonic oscillator has many practical applications, such as in atomic and molecular physics, where it is used to study the energy levels and behavior of particles within atoms and molecules. It is also used in quantum computing and in the development of new technologies, such as lasers and sensors.

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