Calculate the energies of the six lowest states

[acusanelli]

Homework Statement

Suppose that a particle of mass m is confined to move in the x-y plane in a 2-dimensional box of length Lx = L and LY = ½ L. Calculate the energies of the six lowest states.

Homework Equations

not sure to set up this problem?

The Attempt at a Solution

the most I can get is
E = (h^2π^2(n1^2+.5n2^2))/ 2mL^2
E = (h^2π^2)/ mL^2
and I don't think this is right

 

[nickjer]

Why do you have two energy equations? You know how to separate the wavefunction and solve for it in 2 dimensions. What can you say about the total energy after you separated the variables?

 

[acusanelli]

thats because I don't know how to set the problem up, i pulled those equations out of the book and am not sure what I am doing

 

[nickjer]

You should read how the first equation you listed was derived. That way you can reproduce it for your problem.

 

[paranormal]

I would suggest setting up the problem using the Schrödinger equation for a 2-dimensional box potential. This equation represents the allowed energy levels for a particle confined to a specific region, such as the x-y plane in this case. The equation is:

HΨ = EΨ

Where H is the Hamiltonian operator, Ψ is the wavefunction, and E is the energy.

To solve for the energies of the six lowest states, you will need to solve the Schrödinger equation for the lowest six energy eigenvalues. This can be done by using separation of variables and solving the resulting equations for the energy eigenvalues.

Once you have the energy eigenvalues, you can use the equation E = (h^2π^2(n1^2+.5n2^2))/ 2mL^2 to calculate the energies of the six lowest states. Make sure to use the correct values for n1 and n2, which represent the quantum numbers for each dimension.

I would also suggest checking your final answer by comparing it to the expected energies for a 2-dimensional box potential, which are given by:

E = (h^2π^2(n1^2+n2^2))/ 2mL^2

Remember to use the same values for n1 and n2 as you did in the Schrödinger equation.

I hope this helps you solve the problem and obtain the correct energies for the six lowest states. If you need further assistance, don't hesitate to seek guidance from your instructor or consult a textbook or online resources. Good luck!