Quantum harmonic potential problem

In summary, the conversation is about a particle in a harmonic potential and the probability of finding it in the classically excluded region. The participant is trying to find the bounds for the normalization integral and has been advised to use the classical limits for the potential. They are seeking help in determining how much of the wavefunction lies in the excluded region.
  • #1
theWapiti
15
1

Homework Statement



Consider a particle of mass m in a harmonic potential:
gif.latex?V(x)%20%3D%20%5Cfrac%7B1%7D%7B2%7Dm%5Comega%5E2x%5E2.gif


If the particle is in the first excited state (n = 1), what is the probability of finding the
particle in the classically excluded region?

Homework Equations



gif.latex?%5Cint%5E%7B%5Cinfty%7D_%7B%5Csqrt%7B3%7D%7Dx%5E2e%5E%7B-x%5E2%7Ddx%3D0.0495.gif


7B%5Cpartial%20x%5E2%7D%20%3D%20%5Cpsi(%5Cfrac%7Bm%5E2%5Comega%5E2x%5E2-2mE%7D%7B%5Chbar%5E2%7D).gif


The Attempt at a Solution



I sub in
ga%5E2%7D%7B%5Chbar%5E2%7D%0A%5C%5C%0A%5C%5C%0A%5Cbeta%20%3D%20%5Cfrac%7B2mE%7D%7B%5Chbar%5E2%7D.gif


and get a wave function:

7B%5Cfrac%7B1%7D%7B4%7D%7D%5Csqrt%7B2%5Calpha%7Dxe%5E%7B%5Cfrac%7B-%5Calpha%20x%5E2%7D%7B2%7D%7D.gif


But I don't know how to set my bounds for the normalization integral.

I've been advised that the classical limits are:
gif.gif


But I'm still stuck.
 
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  • #2
theWapiti - the classically excluded region is where the potential [itex]V(x)[/itex] exceeds the total energy of the system, which in this case is [itex]\frac{3}{2}\hbar\omega[/itex]. You need to find out how much of your wavefunction lies in this region. The integral given will probably come in useful for doing that.

[I suggest the powers that be move this thread to "Advanced Physics Homework"]
 

FAQ: Quantum harmonic potential problem

1. What is the quantum harmonic potential problem?

The quantum harmonic potential problem is a mathematical model used to study the behavior of a quantum particle in a harmonic potential well. It involves solving the Schrödinger equation to determine the energy levels and wave functions of the particle.

2. How is the quantum harmonic potential problem solved?

The quantum harmonic potential problem is typically solved using the Schrödinger equation and various mathematical techniques, such as the perturbation theory or the variational method. These methods allow for the determination of the energy levels and wave functions of the particle in the potential well.

3. What are the applications of the quantum harmonic potential problem?

The quantum harmonic potential problem has many applications in physics, chemistry, and engineering. It is commonly used to study the behavior of particles in quantum systems, such as atoms and molecules, and to understand the properties of materials at the atomic level. It also has applications in quantum computing and nanotechnology.

4. How does the quantum harmonic potential problem relate to the Heisenberg uncertainty principle?

The quantum harmonic potential problem is related to the Heisenberg uncertainty principle, which states that it is impossible to know both the position and momentum of a particle with absolute certainty. In the case of the quantum harmonic potential problem, the uncertainty principle is reflected in the wave function of the particle, which describes the probability of finding the particle at a certain position and with a certain momentum.

5. What are the differences between the classical and quantum harmonic potential problems?

The classical and quantum harmonic potential problems are fundamentally different. In the classical problem, the particle is treated as a point with well-defined position and momentum, while in the quantum problem, the particle is described by a wave function that represents the probability of finding the particle at a certain position and with a certain momentum. Additionally, the classical problem has a continuous spectrum of energy levels, while the quantum problem has discrete energy levels due to the quantization of energy in quantum mechanics.

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