L, m quantum numbers of 3D oscillator

If so, you can use the orthonormality property of spherical harmonics to find the possible values of l and m for this state. To switch between Cartesian and polar coordinates, you can use the standard equations for conversion.
  • #1
yunseok
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Homework Statement


6 degenerate energy states at E=7/2 h-bar w in isotropic 3D harmonic oscillator.
pick one possible state( for example, (nx,ny,nz)=(1,0,1)), and find possible l, m quantum numbers
you may use orthonormality of spherical harmonics[/B]

Homework Equations


pick one possible state( for example, (nx,ny,nz)=(1,0,1)), and find possible l, m quantum numbers[/B]

The Attempt at a Solution


I tried to understand why the question said 'you may pick (1,0,1), and got it.
But I have no idea with orthonormality. What I know about it is just

double integral 0 to pi and 0 to 2pi (Y(l,m), Y(l',m'))sin(theta)d(theta)d(phi) = delta(mm')delta(ll')

sorry for bad notations.

How can I use this property to get quantum number l, m at (nx, ny, nz)=(1,0,1) ?

also, how can I change quantum numbers from carte to polar and from polar to carte
 

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  • #2
Do you know the functional form of the eigenstate corresponding to (nx,ny,nz)=(1,0,1) in Cartesian coordinate?
 

1. What is the significance of the "L" quantum number in the 3D oscillator?

The "L" quantum number represents the orbital angular momentum of an electron in the 3D oscillator. It determines the shape of the orbital and can have values ranging from 0 to n-1, where n is the principal quantum number.

2. How does the "m" quantum number affect the 3D oscillator?

The "m" quantum number, also known as the magnetic quantum number, determines the orientation of the orbital in space. It can have values ranging from -l to +l, where l is the azimuthal quantum number. The different values of m correspond to different subshells within an energy level.

3. How do the "L" and "m" quantum numbers relate to the energy levels of the 3D oscillator?

The "L" and "m" quantum numbers, along with the principal quantum number, determine the energy levels of the 3D oscillator. The energy of an electron is directly proportional to the value of n, and for a given value of n, it is also affected by the values of L and m. This results in different energy levels and sublevels within an energy level.

4. How does the 3D oscillator differ from the 1D and 2D oscillators?

The 3D oscillator differs from the 1D and 2D oscillators in terms of the number of dimensions it operates in. The 1D oscillator only has one dimension of motion, while the 2D oscillator has two dimensions. The 3D oscillator, on the other hand, operates in all three dimensions, resulting in more complex energy levels and orbital shapes.

5. Can the "L" and "m" quantum numbers have any value in the 3D oscillator?

No, the "L" and "m" quantum numbers must follow certain rules and restrictions in the 3D oscillator. The values of L and m must satisfy the condition that -l ≤ m ≤ l, and the value of L cannot exceed the value of n-1. These restrictions help to determine the unique energy levels and sublevels within an energy level in the 3D oscillator.

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