Quantum Harmonic Oscillator with Additional Potential

In summary, the conversation discusses solving a problem involving Hermite polynomials and a new Hamiltonian with a decaying exponential and additional linear piece. The conversation also mentions making substitutions to reduce the problem to a solvable dimensionless differential equation and asks for suggestions on finding the correct substitution. A hint is given to think about the classical case and the potential and force acting on the particle.
  • #1
Mr_Allod
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16
Homework Statement
Apply and additional potential ##V'(x) = \alpha x## to the standard Hamiltonian of a harmonic oscillator. Find the solutions to the Schrodinger equation.
Relevant Equations
Harmonic Oscillator Hamiltonian: ##H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac {m\omega^2 x^2}{2}##
Hello there, I am trying to solve the above and I'm thinking that the solutions will be Hermite polynomials multiplied by a decaying exponential, much like the standard harmonic oscillator problem. The new Hamiltonian would be like so:

$$H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac {m\omega^2 x^2}{2} + \alpha x$$

Normally one would make the substitutions:

$$y = \sqrt{\frac {m\omega}{\hbar}}$$
$$\epsilon = \frac {2E}{\hbar\omega}$$

This would produce a solvable dimensionless differential equation:
$$\frac {d^2}{dy^2}\psi + (\epsilon-y^2)\psi = 0$$

Now I'm having trouble finding the correct substitution to make to reduce the new problem to a dimensionless one like above. I would appreciate it if someone could give me some suggestions, thank you!
 
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  • #2
Rather think about the Hamiltonian a bit first.

Hint: Think about the classical case. What's the change of the potential due to the additional linear piece? Equivalently you can think about, what it changes for the force acting on the particle and how does it affect the solutions of the classical equations of motion?
 
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FAQ: Quantum Harmonic Oscillator with Additional Potential

1. What is a quantum harmonic oscillator with additional potential?

A quantum harmonic oscillator with additional potential is a physical system that describes the behavior of a particle in a potential energy field that is a combination of a harmonic oscillator potential and an additional potential. It is a common model used in quantum mechanics to study the behavior of particles in various systems such as atoms, molecules, and solid-state materials.

2. What is the significance of studying a quantum harmonic oscillator with additional potential?

Studying a quantum harmonic oscillator with additional potential allows us to better understand the behavior of particles in complex systems. It also helps us to make predictions about the properties and interactions of particles in these systems, which can have practical applications in fields such as materials science, chemistry, and engineering.

3. How is the behavior of a quantum harmonic oscillator with additional potential different from a regular harmonic oscillator?

The behavior of a quantum harmonic oscillator with additional potential is different from a regular harmonic oscillator in that it takes into account the effects of an additional potential energy field. This can result in changes to the energy levels and wavefunctions of the system, leading to different behaviors such as tunneling and anharmonicity.

4. What are some examples of systems that can be described by a quantum harmonic oscillator with additional potential?

Some examples of systems that can be described by a quantum harmonic oscillator with additional potential include diatomic molecules, atoms trapped in optical lattices, and the vibrations of molecules in a crystal lattice. It can also be used to model the motion of electrons in a magnetic field or the vibrations of a molecule in a liquid or gas.

5. How is the Schrödinger equation used to solve for the energy levels and wavefunctions of a quantum harmonic oscillator with additional potential?

The Schrödinger equation, which is the fundamental equation of quantum mechanics, is used to solve for the energy levels and wavefunctions of a quantum harmonic oscillator with additional potential. By solving this equation, we can determine the allowed energy levels of the system and the corresponding wavefunctions, which describe the probability of finding the particle in a certain position. This information can then be used to make predictions about the behavior of the system.

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