Eigenvalue for 1D Quantum Harmonic Oscillator

• theojohn4
In summary, the task is to prove that the given function is an eigenfunction of the Hamiltonian operator for a 1D quantum harmonic oscillator. This involves operating with the Hamiltonian on the function and finding the corresponding eigenvalue. The concept of an eigenfunction means that when the operator is applied to the function, the result is the function multiplied by a constant. The momentum operator is represented by -i\hbar\frac{d}{dx}.
theojohn4

Homework Statement

Show that the following is an eigenfunction of $$\hat{H}_{QHO}$$ and hence find the corresponding eigenvalue:

$$u(q)=A (1-2q^2) e^\frac{-q^2} {2}$$

Homework Equations

Hamiltonian for 1D QHO of mass m
$$\hat{H}_{QHO} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 x^2$$

Not sure about any others

The Attempt at a Solution

I don't know where to start

What means that a function is an eigen-function of some operator? Doesn't this mean that when you act with that operator on that function, the result will be the original function multiplied by a constant (where this constant is the eigenvalue)?

So, try to operate with HQHO on u(q) and see what happens!(the momentum operator is repsresented by -i$\hbar$d/dx)

What is an eigenvalue in the context of a 1D quantum harmonic oscillator?

An eigenvalue in the context of a 1D quantum harmonic oscillator refers to the quantized energy levels of the system. These energy levels are determined by solving the Schrödinger equation for the system and are characterized by a set of discrete values.

How are eigenvalues related to the wave function of a 1D quantum harmonic oscillator?

The eigenvalues of a 1D quantum harmonic oscillator are directly related to the allowed energy states of the system. These energy states are represented by the wave function, which describes the probability amplitude of finding the system in a particular state.

What is the significance of the ground state eigenvalue in a 1D quantum harmonic oscillator?

The ground state eigenvalue in a 1D quantum harmonic oscillator represents the lowest energy state of the system. This state is also known as the zero-point energy state, as it is the minimum amount of energy that the system can possess. The ground state eigenvalue is of particular importance in understanding the behavior of the system.

How do higher energy eigenvalues in a 1D quantum harmonic oscillator differ from the ground state eigenvalue?

Higher energy eigenvalues in a 1D quantum harmonic oscillator correspond to excited states of the system. These states have a higher energy than the ground state and are characterized by a larger number of nodes in the wave function. The difference between these eigenvalues is crucial in understanding the behavior of the system at different energy levels.

What is the relationship between eigenvalues and eigenfunctions in a 1D quantum harmonic oscillator?

The eigenvalues and eigenfunctions in a 1D quantum harmonic oscillator are closely related. The eigenvalues represent the allowed energy levels of the system, while the eigenfunctions represent the corresponding wave functions for these energy levels. Together, they provide a complete description of the quantum state of the system.

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