Is the First Excited State of a Particle in a Rectangular Box Degenerate?

[Squid138]

A particle is confined inside a rectangular box with sides of length a,a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave fuctions have this energy.

The relevant equations are

E=Ex+Ey+Ez
=((hbar^2*pi^2)/2m) +(nx^2)/a^2+(ny^2)/b^2+(nz^2)/c^2

So far I have plugged in the a,a, and 2a into the equation and factored out the a. I am confused at what it means by first excited state. In a hydrogen atom n=2. In this case should each component =2 or should the energy add up to 2?

 

[Tangent87]

Ignore

 

[Squid138]

Wouldn't that imply that there is zero energy in the ground state and is that possible?

 

[Tangent87]

Wouldn't that imply that there is zero energy in the ground state and is that possible?

Ooops sorry I meant (1,1,1) is the ground state and then (1,1,2), (1,2,1) and (2,1,1) will be the first excited state.

 

[Squid138]

How do you know that? Is there something I am not seeing?

 

[Tangent87]

How do you know that? Is there something I am not seeing?

Just look at the energy, if nx,ny,nz are all >=1 then the smallest the energy can be is when they're all equal to 1, that's the ground state. The first excited state is the state of next lowest energy so just look at the expression for energy and decide what nx,ny,nz all have to be, the choice will of course depend on a,b,c.

 

[elduderino]

How do you know that? Is there something I am not seeing?

[What Tangent means] From the boundary conditions you get that n is an integer. You reject negative integers because of they give you no new information. You reject zero because that makes your wave function zero everywhere, and hence makes it non-normalizable. Only things left are n>=1.