Download Quantum Mechanics in 3

Quantum Mechanics in 3- Dimensions
Till this chapter we have solved 1D quantum mechanical problems. In ordet to present a more realistic model we
study the systems in the frame work of the 3 dimensional coordinate system. In this chapter we will discuss 3
problems. 3D infinite well potential, 3D harmonic oscillator (cartesian and spherical coordinate) and Hydrogen
atom.
3D Infinite well potential
Consider a rectangular potential well of sides
Potential
inside the well is zero and outside the well is infinity. Then the
question is how can we obtain wave function and energy of the
particle inside the well.
We are lucky because the wave function outside the well is
because the potential is infinity.
Using the one dimensional analogy one can obtain the wave
functions:
and energy is
For a special case for the cubic potential (a=b=c) we obtain:
The energy and wave function for first few values of the quantum numbers are given by:
1
1
1
3
1
1
2
6
1
2
1
6
2
1
1
6
The last 3 state has the same energy level but different wave function. Or different wave functions corresponding
to the same energy level. This levels is known as degenerate levels.
3D Cartesian harmonic oscillator
Harmonic oscillator potential in 3D is given by
Then the Schrödinger equation takes the form:
Using separation of variable one can easily obtain the solution of the SE by the analogy of the one dimensional
harmonic oscillator problem.
where
are ladder operators as a function of x, y and z. Energy is given by
Table shows energy values of the 3D harmonic oscillator
0
0
0
1
Again the last 3 level are degenerate levels.
0
0
1
0
0
1
0
0
3/2
6
6
6
Spherical coordinate system
Cartesian-spherical and spherical-Cartesian relation can be written as:
And
Then Schrödinger equation can be transformen into Spherical coordinate by a tedious calculation you can obtain:
Then Schrodinger equation takes the form:
Using separation of variables
after some treatments we can obtain:
Therefore
Solution of the angular part yields spherical harmonics
values of it is given in the table. Magnetic quantum number
angular momentum quantum number.
0
0
1
-1
1
0
1
1
it is a special function of physics. Various
takes the values
and is
We can solve the radial equation for harmonic oscillator potential
and we obtain:
where
is Laguerre polynomials and it is again special function of the physics. The quantum numbers
are related by
Energy of the oscillator is given by
Question. Answer the following questions on the harmonic oscillator
a) What are the energy states of 3D Cartesian Harmonic oscillator for each quantum number from 0 to 2.
b) What are the energy states of Harmonic oscillator in spherical coordinate for quantum number from 0 to
2.
Answer:
a)
For 3D Cartesian Harmonic oscillator quantum numbers are
table.
b) For 3D Spherical Harmonic oscillator quantum numbers are
0
3/2
000
000
1
5/2
11-1
100
110
010
111
001
200
110
22-2
101
22-1
011
220
200
221
020
222
001
2
FINAL EXAM UP TO HERE
7/2
. The states given in the
. The states given in the table.