321 Quantum MechanicsUnit 2 Quantum mechanics unit 2 The Schrödinger equation in 3D Infinite...
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Transcript of 321 Quantum MechanicsUnit 2 Quantum mechanics unit 2 The Schrödinger equation in 3D Infinite...
321 Quantum Mechanics Unit 2
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
321 Quantum Mechanics Unit 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
321 Quantum Mechanics Unit 2
3D quantum box
if and if
If then
Quantum numbers,
321 Quantum Mechanics Unit 2
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
|𝑢121|2 |𝑢211|
2 (|𝑢121|¿¿2+|𝑢211|2)/2¿
321 Quantum Mechanics Unit 2
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
321 Quantum Mechanics Unit 2
|𝑢200|2
3D Harmonic Oscillator • Show that the lowest three energy levels are
spherically symmetric
|𝑢110|2 |𝑢020|
2 average
321 Quantum Mechanics Unit 2
Hydrogenic atom• Potential (due to nucleus) is spherically symmetric
Use spherical polar coordinates
nucleus
𝑥
𝑦
𝑧
𝜙
𝜃𝑟
321 Quantum Mechanics Unit 2
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
321 Quantum Mechanics Unit 2
Schrödinger equation in spherical polars
where
and
321 Quantum Mechanics Unit 2
Separation of Schrödinger equation • Radial equation
• equation
• represents the angular dependence of the wavefunction in any spherically symmetric potential
321 Quantum Mechanics Unit 2