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Now that we have redefined our Schrödinger equation in 3 dimensions, let us see how this effects the quantum harmonic oscillator (QHO) problem we solved earlier. The 2D parabolic well will now turn into a 3D paraboloid.
To solve this equation of the well, we are going to make our separation of variables approximation for a standing wave (just like we did for the free particle):
We are now in a position to solve the Schrödinger equation in equation 1:Schrödinger
In order to make sure everyone is following, let us review some key steps below:
5: The 3-dimensional Schrödinger equation
5 to 6: Plug in our separation of xyz ansatz
6 to 7: Distribute out the RHS of the equation. Note that the d/dx derivative only acts on psi(x) not psi(y) or psi(z)
7 to 8: Divide both sides by psi(x) psi(y) psi(z). The total energy of the particle is constant
In equation 8, if the particle does not move in the x or y directions at all, the purple and blue terms are zero. This would mean that the green term, regardless of if I move in the z direction, is a constant. We can make the same argument for the purple and blue terms as well. Each colored term is just a constant.
Additionally, one might notice that the purple, blue, and green terms in equation 6 all independently look like the 1-dimensional Hamiltonian equation (for the x, y, and z dimension respectively). In 1-dimension they were also equal to their own constant. We can, just for simplicity, call this constant E_x, E_y, and E_z:
5: The 3-dimensional Schrödinger equation
5 to 6: Plug in our separation of xyz ansatz
6 to 7: Distribute out the RHS of the equation. Note that the d/dx derivative only acts on psi(x) not psi(y) or psi(z)
7 to 8: Divide both sides by psi(x) psi(y) psi(z). The total energy of the particle is constant
In equation 8, if the particle does not move in the x or y directions at all, the purple and blue terms are zero. This would mean that the green term, regardless of if I move in the z direction, is a constant. We can make the same argument for the purple and blue terms as well. Each colored term is just a constant.
Additionally, one might notice that the purple, blue, and green terms in equation 6 all independently look like the 1-dimensional Hamiltonian equation (for the x, y, and z dimension respectively). In 1-dimension they were also equal to their own constant. We can, just for simplicity, call this constant E_x, E_y, and E_z:
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In order to make sure everyone is following, let us review some key steps below:
7: Take the constant 'E' and break it up into three different constants
We then separated out the constants into 3 different 1D equations (of which we have already solved).
Our equations exactly match that of the 1-dimensional quantum harmonic oscillator. We have already solved this solution (using the brute force method and Hermite polynomials; see page for more details), so I won't go through all that hectic math (because it is the exact same).
We can also add up all the individual energies to find the total energy below:
7: Take the constant 'E' and break it up into three different constants
We then separated out the constants into 3 different 1D equations (of which we have already solved).
Our equations exactly match that of the 1-dimensional quantum harmonic oscillator. We have already solved this solution (using the brute force method and Hermite polynomials; see page for more details), so I won't go through all that hectic math (because it is the exact same).
We can also add up all the individual energies to find the total energy below:
Just like in 1-dimension, the energy states are separated in 'h_bar * w' increments (like a ladder). However, it is important to now realize that, unlike in 1-dimension, in 3-dimensions we CAN have degenerates states. An example of this can be seen below:
The only way to break the degeneracy is to break the symmetry of the problem (giving each dimension its own angular frequency w). If this occurs, then the energy reduces to the form below:
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