1. Nanoelectronics Chapter 3 Quantum Mechanics of Electrons 1Q.Li@Physics.WHU@2015.3

2. STM image of atomic “quantum corral” Q.Li@Physics.WHU@2015.32 Atoms form a quantum corral to confine the surface state electrons.

3. 3.1 General Postulates of Quantum Mechanics Q.Li@Physics.WHU@2015.33

4. 3.1 General Postulates of Quantum Mechanics 3.1.1 Operators Q.Li@Physics.WHU@2015.34

5. 3.1.2 Eigenvalues and Eigenfunctions Q.Li@Physics.WHU@2015.35

6. 3.1.3 Hermitian Operator Hermitian operators have real eigenvalues. Their eigenfunctions form an orthogonal, complete set of functions. Q.Li@Physics.WHU@2015.36 (if normalized)

7. 3.1.4 Operators for Quantum Mechanics Momentum operator Energy operator Q.Li@Physics.WHU@2015.37

8. 3.1.4 Operators for Quantum Mechanics Position operator Q.Li@Physics.WHU@2015.38 The eigenfunction is

9. 3.1.4 Operators for Quantum Mechanics Commutation and the Uncertainty principle Q.Li@Physics.WHU@2015.39 α and β operators are commute The difference operator: is commutor  So one cannot measure x and p x (along x-axis) with arbitrary precision They are not commute!

10. 3.1.4 Operators for Quantum Mechanics Uncertainty principle Q.Li@Physics.WHU@2015.310 So one can measure x and p y (along y-axis) with arbitrary precision

11. 3.1.5 Measurement Probability Postulate 3: The mean value of an observable is the expectation value of the corresponding operator. Postulate 4: Q.Li@Physics.WHU@2015.311

12. 3.2 Time-independent Schrodinger’s Equation Q.Li@Physics.WHU@2015.312 Separation of variables

13. 3.2.1 Boundary Conditions on Wavefunction Q.Li@Physics.WHU@2015.313 Consider a one-dimensional space with electrons constrained in 0<x<L

14. Evidence for existence of electron wave Q.Li@Physics.WHU@2015.314

15. 3.3 Analogies between Quantum Mechanics and Classical Electromagnetics Maxwell’s equations: Q.Li@Physics.WHU@2015.315 comparison

16. 3.4 Probabilistic current density Q.Li@Physics.WHU@2015.316

17. 3.5 Multiple Particle Systems State function Joint probability of finding particle 1 in d 3 r 1 point r 1 and finding particle 2 in d 3 r 2 of point r 2 State function obeys Q.Li@Physics.WHU@2015.317

18. 3.5 Multiple Particle Systems Hamiltonian: Example: two charged particles: Q.Li@Physics.WHU@2015.318

19. 3.6 Spin and Angular Momentum Lorentz force If the particle has a net magnetic moment µ, passing through a magnetic field B Angular momentum: Spin is a purely quantum phenomenon that cannot be understood by appealing to everyday experience. (it is not rotating by its own axis.) Q.Li@Physics.WHU@2015.319

20. 3.7 Main Points Meaning of state function Probability of finding particles at a given space Probability of measuring certain observable Operators, eigenvalues and eigenfunctions Important quantum operators Mean of an observable Time-dependent/independents Schrodinger equations Probabilistic current density Multiple particle systems Q.Li@Physics.WHU@2015.320

21. 3.8 Problems 1, 3, 8, 9, 15 Q.Li@Physics.WHU@2015.321