Show that the two wave functions are eigenfunction

In summary, the conversation discusses the dimensionless harmonic oscillator Hamiltonian and the two wave functions ψ0(x)=e-x2/2 and ψ1(x)=xe-x2/2 that are eigenfunctions of the Hamiltonian with eigenvalues ½ and 3/2, respectively. The conversation also mentions finding the coefficient a for ψ2(x)=(1+ax2)e-x2/2 to be orthogonal to ψ0(x), and then proving that ψ2(x) is an eigenfunction of H with eigenvalue 5/2. The method involves applying definitions and making comparisons between the eigenvalues.
  • #1
gfxroad
20
0

Homework Statement


Consider the dimensionless harmonic oscillator Hamiltonian
H=½ P2+½ X2, P=-i d/dx.
  1. Show that the two wave functions ψ0(x)=e-x2/2 and ψ1(x)=xe-x2/2 are eigenfunction of H with eigenvalues ½ and 3/2, respectively.
  2. Find the value of the coefficient a such that ψ2(x)=(1+ax2)e-x2/2 is orthogonal to ψ0(x). Then show that ψ2(x) is an eigenfunction of H with eigenvalue 5/2.

The Attempt at a Solution


For orthogonality the wave function product must equal to zero, and for eigenfunction we take the second derivative for both wave functions and make a comparison between the eigenvalues.
But I can't finalise the problem, so I appreciate any help in advance.
 
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  • #2
Please show working.
Your method is sort of OK.
It's a bit more than just taking the product or the second derivative.
You have to apply the definitions.
 

FAQ: Show that the two wave functions are eigenfunction

What does it mean for two wave functions to be eigenfunctions?

Two wave functions are considered eigenfunctions if one can be expressed as a scalar multiple of the other. In other words, one wave function is a constant multiple of the other, with the same shape and frequency.

How do you show that two wave functions are eigenfunctions?

To show that two wave functions are eigenfunctions, one must demonstrate that they satisfy the eigenvalue equation. This involves multiplying one function by a scalar value and comparing it to the other function. If they are equal, then they are eigenfunctions.

Why is it important to determine if two wave functions are eigenfunctions?

Determining if two wave functions are eigenfunctions is important in quantum mechanics, as it allows for the simplification of complex equations and makes it easier to solve for the wave function of a system. It also helps to identify the energy levels of a system.

Can two different wave functions be eigenfunctions?

Yes, two different wave functions can be eigenfunctions as long as one can be expressed as a scalar multiple of the other. This means that although the functions may have different shapes and frequencies, they still have the same underlying structure and can be considered eigenfunctions.

What are some examples of wave functions that are commonly found to be eigenfunctions?

Some common examples of eigenfunctions include sine waves, cosine waves, and exponential functions. These functions are often used to describe the behavior of particles in quantum mechanics and can be expressed as scalar multiples of each other.

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