5. Quantum Theory

1. 5. Quantum Theory 5.0. Wave Mechanics 5.1. The Hilbert Space of State Vectors 5.2. Operators and Observable Quantities 5.3. Spacetime Translations and Properties of Operators 5.4. Quantization of a Classical System 5.5. An Example: the One-Dimensional Harmonic Oscillator

2. 5.0. Wave Mechanics • Cornerstones of quantum theory: • Particle-wave duality • Principle of uncertainty Planck’s constant History: Planck: Empirical fix for black body radiation. Einstein: Photo-electric effect → particle-like aspect of “waves”. de Broglie: Particle-wave duality. Thomson & Davisson: Diffraction of electrons by a crystal lattice. Schrodinger: Wave mechanics.

3. Wave mechanics State of a “particle” is represented by a (complex) wave function Ψ( x, t ). Probability of finding the particle in d 3x about x at t = P = probability density if Ψ( x, t ) is called normalized if c = 1. Pd 3x = relative probability if Ψ cannot be normalized. E.g., free particle: (1st) quantization: → r-representation:

4. Hamiltonian : → Time-dependent Schrodinger equation : c.f. Hamilton-Jacobi equation Time-independent Schrodinger equation :

5. 5.1. The Hilbert Space of State Vectors Specification of a physical state: Maximal set of observables M = { A, B, C, …}. → Pure quantum state specified by values { a, b, c, … } assumed by S. Assumption: Every possible instantaneous state of a system can be represented by a ray (direction) in a Hilbert space. (see Appendix A.3) Hilbert spaces are complex linear vector spaces with possibly infinite dimensions. • φ | = 1-form dual to | φ  = | φ † Inner product is sesquilinear :  α, β  C, → Norm / length / magnitude of | φ is

6. | φ is normalized if Maximal set M = { A, B, C, …} → pure states are given by | a, b, c, … . Probability of measuring values { a, b, c, … } from a state | Ψ  is →  orthonormality → → Completeness  else

7. If a takes on continuous values, Example: 1-particle system with x as maximal set. orthonormality completeness Ψ (x) = wave function

8. 5.2. Operators and Observable Quantities Operator: E.g., identity operator: Linear operators:  α, β C Observables are represented by linear Hermitian operators. Let the maximal set of observables be M = { A, B, C, …}. If we choose | a,b,c,…  as basis vectors, then the operators are defined as Eigen-equations eigenvalues eigenvectors …

9. Expectation value of A: AdjointA† of A is defined as: → A is self-adjoint / hermitian if

10. Consider 2 eigenstates, | a1 and | a2, of A. → If A is hermitian, Hence, → Eigenvalues of a hermitian operator are all real. → Eigenstates belonging to different eigenvalues of a hermitian operator are orthogonal.

11. Algebraic Operations between Operators Addition: Product: Commutator: Analytic functions of A are defined by Taylor series. Ex. 5.7 Caution: unless If B is hermitian, then → ( A is unitary )

12. 5.3. Spacetime Translations and Properties of Operators Time Evolution: Schrodinger Picture Each vector in Hilbert space represents an instantaneous state of system. U is the time evolution operator Normalization remains unchanged : → U is unitary Setting we can write where If H is time-independent, → H = Hamiltonian c.f. Liouville eq.

13. Heisenberg Picture | Ψ (t)  is not observable. Observable: for time independent H → A is conserved if it commutes with H. H is conserved. Classical mechanics: Possible rule: ( Not always correct ) Example: Canonical commutation relations

14. Alternative derivation: Classical translational generator: → All components of x or p should be simultaneously measurable → → Consider → → See Ex.5.3 for higher order terms →  H independent of x (translational invariant)

15. Classical mechanics: Conserved quantity ~ L invariant under corresponding symmetry transformation Quantum mechanics: Conserved operator = generator of symmetry transformation Example: Angular Momentum

16. 5.4. Quantization of a Classical System Canonical quantization scheme: Classical Quantum Schrodinger Heisenberg • Difficulties: • Generalized coordinates may not work. • Remedy: Stick with Cartesian coordinates. • Ambiguity. E.g., AB when Possible remedy: use Remedy: use pi . • Constrainted or EM systems with Reminder: velocity is ill-defined in QM.

17. Wave functions: x-representation (Taylor series) → Many bodies:

18. 5.5. An Example: the One-Dimensional Harmonic Oscillator Some common choices of maximal sets of observables: { x } : coordinate (x-) representation { p } : momentum (p-) representation { E } : energy (E-) representation { n } : number (n-) representation n-representation Basis vectors are eigenstates of number operator: Lowering (annihilation) operator: Raising (creation) operator: Canonical quantization:

19. → → → … → → Setting cn and bn real gives

20. Exercise: Show that if n is restricted to the values 0 and 1, then the commutator relation must be replaced by the anti-commutator relation

21. For the harmonic oscillator, if we set then → Hence, basis { | n } is also basis of the E-representation. The nth excited state contains n vibrons, each of energy  ω. → →

22. x-representation: → where →  so that with Using one can show where and

23. The x- and p- representations are Fourier transforms of each other. where so that Similarly where In practice, ψ (x) is easier to obtain by solving the Schrodinger eq. with appropriate B.C.s

24. Let V(x) → 0 as |x| → . For E < 0, ψn(x) is a bound state with discrete eigen-energies εn. For E > 0, ψk(x) is a scattering state with continuous energy spectrum ε(k). However, scattering problems are better described in terms of the S matrix, scattering cross sections, or phase-shifts.