Simple harmonic oscillator Hamiltonian

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The time-independent Schrödinger equation for a simple harmonic oscillator can be written as $$\hat{H}|\psi>=E_n|\psi>$$ where $$\hat{H}=\frac{1}{2}mw^2\hat{x}^2+\frac{\hat{p}^2}{2m}$$. This can be shown by working backwards from the equation $$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)|\psi>=E_n|\psi>$$ In summary, the time-independent Schrödinger equation for a simple harmonic oscillator can be written as $$\hat{H}|\psi>=E_n|\psi>$$
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Homework Statement
Relevant Equations

We show by working backwards
$$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)=\hbar w \Big(\frac{mw}{2\hbar}(\hat{x}+\frac{i}{mw}\hat{p})(\hat{x}-\frac{i}{mw}\hat{p})+\frac{1}{2}\Big)$$
$$=\Big(\frac{mw^2}{2}(\hat{x}^2+\frac{i}{mw}[\hat{p},\hat{x}]+\frac{\hat{p}^2}{m^2w^2})+\frac{1}{2}\Big)=\frac{1}{2}mw^2\hat{x}^2-\frac{\hbar w}{2}+\frac{\hat{p}^2}{2m}+\frac{\hbar w}{2}$$
$$=\frac{1}{2}mw^2\hat{x}^2+\frac{\hat{p}^2}{2m}=\hat{H}$$
This shows the time-independent Schrödinger equation for the simple harmonic oscillator can be written as
$$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)|\psi>=E_n|\psi>$$

Last edited:
Delta2 and PeroK
I think you confuse the greek letter ##\omega## with the latin letter ##w## ...
Ssnow

1. What is a simple harmonic oscillator Hamiltonian?

The simple harmonic oscillator Hamiltonian is a mathematical representation of a system that exhibits simple harmonic motion, which is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium. It is commonly used in physics and engineering to model systems such as springs and pendulums.

2. What is the equation for the simple harmonic oscillator Hamiltonian?

The equation for the simple harmonic oscillator Hamiltonian is H = (1/2)kx^2 + (1/2)mv^2, where k is the spring constant, x is the displacement from equilibrium, m is the mass of the object, and v is the velocity. This equation represents the total energy of the system, which is the sum of the potential energy and kinetic energy.

3. How is the simple harmonic oscillator Hamiltonian used in quantum mechanics?

In quantum mechanics, the simple harmonic oscillator Hamiltonian is used to describe the behavior of quantum mechanical systems, such as atoms and molecules. It is used to calculate the energy levels and wavefunctions of these systems, which can then be used to make predictions about their behavior.

4. What is the significance of the simple harmonic oscillator Hamiltonian?

The simple harmonic oscillator Hamiltonian is significant because it is a fundamental model that can be applied to a wide range of physical systems. It is also one of the few systems in classical mechanics that can be solved exactly, making it an important tool for studying more complex systems.

5. How does the simple harmonic oscillator Hamiltonian relate to the Schrödinger equation?

The simple harmonic oscillator Hamiltonian is used as the potential energy term in the Schrödinger equation, which is the fundamental equation of quantum mechanics. This allows for the calculation of the wavefunction and energy levels of a quantum system, making the simple harmonic oscillator Hamiltonian an essential tool in the field of quantum mechanics.

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