# I Harmonic Oscillator in 3D, different values on x, y and z

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1. Sep 12, 2016

### Ofinns

Hi,

For a harmonic oscillator in 3D the energy level becomes En = hw(n+3/2) (Note: h = h_bar and n = nx+ny+nz) If I then want the 1st excited state it could be (1,0,0), (0,1,0) and (0,0,1) for x, y and z.

But what happens if for example y has a different value from the beginning? Like this: V(x,y,z) = 1/2mw2(x2+4y2+z2) and for this decide the energy level AND degeneracy for the 1st excited state. I can only find simple examples when x, y and z are equal and 1.

Best regards

2. Sep 12, 2016

### Demystifier

In such a more general case you have
$$E_{n_1n_2n_3}=\hbar \omega_1 \left( n_1+\frac{1}{2} \right) + \hbar \omega_2 \left( n_2+\frac{1}{2} \right) + \hbar \omega_3 \left( n_3+\frac{1}{2} \right)$$

3. Sep 12, 2016

### Ofinns

Can you elaborate on that? Is 4y2 just n2 here? And in that case you will get three different energy values:

E100 = 3hw1/2
E010 =6hw2
E001 =3hw3/2

Which one is the 1st excited state? Is it E010?

4. Sep 12, 2016

Staff Emeritus
No, it enters in as the frequency.

5. Sep 12, 2016

### Demystifier

$$\omega_1=w$$
$$\omega_2=2w$$
$$\omega_3=w$$
Therefore
$$E_{000}=2\hbar w$$
$$E_{100}=E_{001}=3\hbar w$$
$$E_{010}=4\hbar w$$
Hence the first excited states are $E_{100}=E_{001}$.

6. Sep 12, 2016

### Ofinns

Thank you, now I understand that part.

What will the degeneracy become for the 1st excited state then? Can I use the same formula gn = 1/2(n+1)(n+2) for this case?

7. Sep 12, 2016

### Demystifier

It's 2.

No.

8. Sep 12, 2016

### Ofinns

Why is it 2? What formula do you use to calculate that? (Sorry for all the questions..)

9. Sep 12, 2016

### Demystifier

It follows from the last line of post #5. There you see that there are 2 "first excited states" with equal energies. Hence the degeneracy of first excited state is 2.

10. Sep 12, 2016

### Ofinns

Oh! Thank you so much for the answers, this has been bugging me for a while now.

Best regards

11. Sep 22, 2016

### Ofinns

Late questions.. but why is w2=2w and not 4w?

12. Sep 22, 2016

### Demystifier

Because, by definition,
$$V(x)=\frac{1}{2}m\omega^2 x^2$$

13. Sep 22, 2016

### Ofinns

Right, of course. Thank you.