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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.