321 Quantum Mechanics Unit 1

Quantum mechanics unit 1• Foundations of QM

• Photoelectric effect, Compton effect, Matter waves

• The uncertainty principle

• The Schrödinger eqn. in 1D

• Square well potentials and 1D tunnelling

• The harmonic oscillator

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321 Quantum Mechanics Unit 1

Last time• Heisenberg uncertainty principle

• Matter waves obey the Schrödinger equation

• The wavefunction contains all the physical information about the system.

• dx

• To be physically meaningful the wavefunction must obey some constraints

321 Quantum Mechanics Unit 1

Time independent Schrödinger equation

eigenfunction

eigenvalue

321 Quantum Mechanics Unit 1

Constraints• The wavefunction and its first derivative must be:

• Single valued

• Finite

• Continuous

321 Quantum Mechanics Unit 1

Time independent Schrödinger equation• For a given quantum system –

• solve S.E. to find allowed energy levels, E• find and probability distribution

• Examples: • Infinite square well• Finite square well• Square barrier• Harmonic oscillator potential

321 Quantum Mechanics Unit 1

1. An electron is confined in a box with half-width Å. Calculate the energy separation between ground state and first excited state. What is the wavelength of the photon which would cause an electron to be excited from the ground state to the first exited state?

2. A grain of salt, kg, is trapped in a box with m. Calculate the ground state energy. What is the excited state number, if the energy is J (corresponding to at K)?

Infinite square well example 𝐸𝑛=ℏ2 𝜋 2𝑛2

8𝑚𝑎2

eV, Å

eV,

321 Quantum Mechanics Unit 1

Finite square well

𝑉 0

−𝑎 𝑎𝑉=0

1. Write down solutions to S.E. in each region2. Apply the boundary conditions to find any unknown constants

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321 Quantum Mechanics Unit 1

𝑉 0=25ℏ2

2𝑚𝑎2

Graphical solution:

𝑘0𝑎=5

Even parity states

𝑥=𝑘𝑎=1.3 𝑥=3 .8

321 Quantum Mechanics Unit 1

𝑉 0=25ℏ2

2𝑚𝑎2

𝑘0𝑎=5

Odd parity states

𝑥=𝑘𝑎=2.6 𝑥=4.9

Graphical solution:

321 Quantum Mechanics Unit 1

Compare infinite to finite wellWell half width, Å, Finite well depth, eV

Infinite well eV eV eV…

Finite well eV eV eV eV