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Dr Rashid, 2020Jashore University of Science and Technology
Three-Dimensional
Infinite-Potential Well
Dr Mohammad Abdur Rashid
Dr Rashid, 2020Jashore University of Science and Technology
Infinite Square-Well Potential
Dr Rashid, 2020Jashore University of Science and Technology
Two-Dimensional Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
Separation of Variables Technique
The wave function can be written as a product of
functions, one depending only on x and the other
only on y:
The total energy is given by
Dr Rashid, 2020Jashore University of Science and Technology
Separation of Variables Technique
Substituting these results to the Schrödinger Equation
Dr Rashid, 2020Jashore University of Science and Technology
Separation of Variables Technique
The first term on the left is only dependent on x, similarly the
second term on the left is only dependent on y and their sum is a
constant. Therefore, we can write
Dr Rashid, 2020Jashore University of Science and Technology
Two-Dimensional Infinite-Potential Well
Finally we have two ordinary (that is, single-variable)
differential equations to solve
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
Schrödinger Equation in 3D
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
The solution of the Schrödinger equation as in 2D
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
In order to find the energies, we first need to take the
appropriate derivatives of the wave function. We do
this first for the variable x.
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
The derivatives for y and z are similar. With these derivatives
we have
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
The normalized wave function
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
For the ground state
3D Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
What is the energy of the first excited state?
3D Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
3D Infinite-Potential Well
Dr Rashid, 2020Jashore University of Science and Technology
In physics we say that a given state is degenerate whenthere is more than one wave function for a given energy.
In our case all three possible wave functions for the firstexcited state have the same energy. The degeneracy inthis case is a result of the symmetry of the cube.
If the box had sides of three different lengths, we say thedegeneracy is removed, because the three quantumnumbers in different orders (211, 121, 112) would resultin three different energies.
Degenerate State
Dr Rashid, 2020Jashore University of Science and Technology
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