How Do Eigenstates and Energies Change with Potential in Quantum Mechanics?

[EEnerd]

Homework Statement

in the 3rd problem consider a particle with mass m and of potential energy
V(x,y,z)=( (mω^2)/2) * [(1+ (2λ/3))*(x^2 +y^2) + (1- (4λ/3))*z^2]
where ω , λ are constants ω≥0 , 0≤λ≤3/4
what are the eigenstates of the Hamiltonion and the corresponding energies
whats the degeneracy of the ground state λ=0 and the first state λ=3/4

Homework Equations

The Attempt at a Solution

ok if λ=0 i know that v= 1/2 ω(^2) m (x^2+y^2+z^2) which means the H=Hx+Hy+Hz, and that means E=(nx +ny+nz+3/2) homework and the ground state u have (1.0.0) and (0.1.0). and (0.0.1) and E= 5/2 homework
for λ=3/4 V=1/2 mw^2(3/2)(x^2+y^2) so
v=3/2 1/2 mw^2(x^2+y^2)
so H=Hx+Hy=(nx+ny+1)3/2 h w and the ground state degeneracy is (1.0) (0.1) ?! ( i know i made a lot of mistakes :P, but i am EE major and this class is supposed to be introductory quantum)

i am not sure about the eigen states, but it looks like a harmonic oscillator so its ground state it should be ψ=N0e^-1/2 ζ ?!

 

[DrClaude]

what are the eigenstates of the Hamiltonion and the corresponding energies
whats the degeneracy of the ground state λ=0 and the first state λ=3/4

This is not clear. Can you check the exact formulation of the problem?

ok if λ=0 i know that v= 1/2 ω(^2) m (x^2+y^2+z^2) which means the H=Hx+Hy+Hz, and that means E=(nx +ny+nz+3/2) homework and the ground state u have (1.0.0) and (0.1.0). and (0.0.1) and E= 5/2 homework

These are not the correct quantum numbers for the ground state.

for λ=3/4 V=1/2 mw^2(3/2)(x^2+y^2) so
v=3/2 1/2 mw^2(x^2+y^2)
so H=Hx+Hy=(nx+ny+1)3/2 h w and the ground state degeneracy is (1.0) (0.1) ?!

Do you know what degeneracy mean? And don't forget about the $z$ axis.

i am not sure about the eigen states, but it looks like a harmonic oscillator so its ground state it should be ψ=N0e^-1/2 ζ ?!

What is ζ?

 

[EEnerd]

This is not clear. Can you check the exact formulation of the problem?


These are not the correct quantum numbers for the ground state.


Do you know what degeneracy mean? And don't forget about the $z$ axis.


What is ζ?

ok the questions says find the eigenstates and the corresponding energies , and the degeneracy is when many states or multiple states share the same energy value right?!

 

[EEnerd]

ok i see what i did, for ground state n=0 and that means (0,0,0) and E=3/2hw, and for the first state when y=3/4 i will get (0.1.0) , (1.0.0) ?! and E will be 3hw?!

 

[DrClaude]

and the degeneracy is when many states or multiple states share the same energy value right?!

Yes, but when the question asks for the degeneracy of a state, the answer is a single number (or formula).

 

[EEnerd]

Yes, but when the question asks for the degeneracy of a state, the answer is a single number (or formula).

oh so in ground state when we have n=0 we got degeneracy=0 or 1?! cause there is only one state !
and when we have n=1 , we have degeneracy =3 ?

 

[EEnerd]

Thanks a lot for the help, the website is actually better than our school TAs, :)

 

[DrClaude]

oh so in ground state when we have n=0 we got degeneracy=0 or 1?!
cause there is only one state !

You would call that a degeneracy of 1 (even though it is not degenerate!).

and when we have n=1 , we have degeneracy =3 ?

[/QUOTE]
Correct.