Quantum mechanics unit 1

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 www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures

2. 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

3. Time independent Schrödinger equation eigenfunction eigenvalue

4. Constraints • The wavefunction and its first derivative must be: • Single valued • Finite • Continuous

5. 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

6. Infinite square well example • 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? • 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)? eV, Å eV,

7. Finite square well Write down solutions to S.E. in each region Apply the boundary conditions to find any unknown constants www2.le.ac.uk/departments/physics/people/academic-staff/mr6/lectures

8. Graphical solution: Even parity states

9. Graphical solution: Odd parity states

10. Compare infinite to finite well Well half width, Å, Finite well depth, eV Infinite well eV eV eV … Finite well eV eV eV eV