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Quantum mechanics unit 2

Quantum mechanics unit 2. The Schrödinger equation in 3D I nfinite quantum box in 3D & 3D harmonic oscillator The Hydrogen atom Schrödinger equation in spherical polar coordinates Solution by separation of variables Angular quantum numbers Radial equation and principal quantum numbers

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Quantum mechanics unit 2

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  1. Quantum mechanics unit 2 • The Schrödinger equation in 3D • Infinite quantum box in 3D & 3D harmonic oscillator • The Hydrogen atom • Schrödinger equation in spherical polar coordinates • Solution by separation of variables • Angular quantum numbers • Radial equation and principal quantum numbers • Hydrogen-like atoms Rae – Chapter 3 www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures

  2. Last time • Time independent Schrödinger equation in 3D • u must be normalised, u and its spatial derivatives must be finite, continuous and single valued • If then and the 3D S.E. separates into three 1D Schrödinger equations - obtain 3 different quantum numbers, one for each degree of freedom • Time independent wavefunctions also called stationary states

  3. 3D quantum box if and if If then Quantum numbers,

  4. Degeneracy • States are degenerate if energies are equal, eg. • Degree of degeneracy is equal to the number of linearly independent states (wavefunctions) per energy level • Degeneracy related to symmetry

  5. 3D Harmonic Oscillator • Calculate the energy and degeneracies of the two lowest energy levels Ground state is undegenerate, or has degeneracy 1 1st excited state is 3-fold degenerate 2nd excited state has degeneracy 6 - don’t forget for a harmonic oscillator

  6. 3D Harmonic Oscillator • Show that the lowest three energy levels are spherically symmetric average

  7. Hydrogenic atom • Potential (due to nucleus) is spherically symmetric Use spherical polar coordinates nucleus

  8. Hydrogenic atom • so, can separate the wavefunction • Solve separately for • continuous, finite, single valued, = 1 • Expect 3 quantum numbers - as 3 degrees of freedom • Expect as because state is bound • Expect (result from Bohr’s theory) • Expect degenerate excited states

  9. Schrödinger equation in spherical polars where and

  10. Separation of Schrödinger equation • Radial equation • equation • represents the angular dependence of the wavefunction in any spherically symmetric potential

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