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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 5.0. Wave Mechanics Cornerstones of quantum theory: • Particle-wave duality • Principle of uncertainty E h h 2 h p h 6.6256 1034 J s 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. k Planck’s constant Wave mechanics State of a “particle” is represented by a (complex) wave function Ψ( x, t ). Probability of finding the particle in d P = probability density if 3x about x at t = P x, t d x c x, t 3 P x, t d x c x, t 3 2 d 3x 1 Ψ( x, t ) is called normalized if c = 1. P d 3x = relative probability if Ψ cannot be normalized. E.g., free particle: x, t exp i k x t (1st) quantization: A A x, p r-representation: → xˆ , pˆ i Aˆ A xˆ , pˆ xˆ x pˆ i 2 d 3x Hamiltonian : 1 2 H p V x 2m Time-dependent Schrodinger equation : i Hˆ → 2 2m 2 V x c.f. Hamilton-Jacobi equation x, t Hˆ x, t t 2 2 i x, t V x x, t t 2 m Time-independent Schrodinger equation : Hˆ x, t E x, t 2 2 V x x, t E x, t 2m Ĥ i t 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 * 1 | 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 P a, b, c, | a, b, c, → P a, b, c, a, b, c, a, b, c, P a, b, c, a, b, c, | 1 → a ,b , c , abc 2 | aa bb cc aa bb cc orthonormality a, b, c, 1 a, b, c, a ,b , c , abc a, b, c, a, b, c, a ,b , c , a, b, c, a , b, c , a, b, c, a, b, c, 1 a ,b , c , → Completeness a ,b , c , 1 else P a, b, c, | a, b, c, 2 a a a a If a takes on continuous values, a da Example: 1-particle system with x as maximal set. x x x x 3 d x x d 3x x orthonormality x 1 x completeness d 3x x x P x | x 2 Ψ (x) = wave function 5.2. Operator: Operators and Observable Quantities Ô Iˆ E.g., identity operator: Ô Linear operators: Oˆ Oˆ Oˆ 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 Mˆ Aˆ , Bˆ , Cˆ , are defined as Aˆ a, b, c, a a, b, c, Bˆ a, b, c, b a, b, c, Cˆ a, b, c, c a, b, c, Eigen-equations eigenvalues eigenvectors … P a a, b, c, 2 b, c, Expectation value of A: A a P a a a, b, c, a, b, c, a, b, c, a Aˆ a, b, c, a, b, c, a , b, c,  Adjoint → A† of A is defined as: Aˆ † Aˆ A is self-adjoint / hermitian if Aˆ † Aˆ † Aˆ † Aˆ †  * Consider 2 eigenstates, | a1 and | a2 , of A.  a1 a1 a1 →  a2 a2 a2 a2 Aˆ a1 a1 a2 If A is hermitian, Hence, a2 a1 a a 1 → a2 Aˆ † a1 a2* a2 a1 * 2 a1 0 a2 a1 a1* is real Eigenvalues of a hermitian operator are all real. a1 a2 → a2 a1 0 Eigenstates belonging to different eigenvalues of a hermitian operator are orthogonal. a1 Algebraic Operations between Operators Addition: Aˆ Bˆ Product: Aˆ Bˆ Aˆ Bˆ Aˆ Bˆ Aˆ , Bˆ Aˆ Bˆ Bˆ Aˆ Commutator: Analytic functions of A are defined by Taylor series. Aˆ e Bˆ n ˆn 1 B 1 Bˆ 2 Bˆ 2 n! n 0 Caution: ˆ ˆ ˆ ˆ eB eC e B C If B is hermitian, then 2 Bˆ , Cˆ 0 unless ˆ Aˆ ei B → ˆ 1 Aˆ † e i B  Ex. 5.7 ( A is unitary ) 5.3. Spacetime Translations and Properties of Operators Time Evolution: Schrodinger Picture Each vector in Hilbert space represents an instantaneous state of system. t Uˆ t 0 U is the time evolution operator Normalization remains unchanged : t t 0 0 → Setting Uˆ † t Uˆ t Iˆ Uˆ 0 Iˆ 0 Uˆ † t Uˆ t 0 U is unitary we can write ˆ Uˆ t ei H t where Hˆ † Hˆ If H is time-independent, i d d t i Uˆ t 0 dt dt 1 → Hˆ Ĥ ˆ Hˆ ei H t 0 H = Hamiltonian Hˆ Uˆ t 0 Hˆ t c.f. Liouville eq. | Ψ (t) is not observable. Heisenberg Picture A t t Aˆ t Observable: 0 Aˆ t ˆ ˆ t Aˆ t Uˆ † t AU e → i i ˆ Ht Aˆ e ˆ ˆ t 0 Uˆ † t AU 0 i ˆ Ht for time independent H d ˆ A t Aˆ t , Hˆ dt A is conserved if it commutes with H. Classical mechanics: Possible rule: 0 d A t A , H dt Aˆ , Bˆ ~ i A, B P H is conserved. P ( Not always correct ) Example: Canonical commutation relations xˆ , pˆ i a xˆ , xˆ pˆ , pˆ 0 Alternative derivation: P i P, Classical translational generator: 1 1 Pˆ Pˆ pˆ j → P H i H , P T ˆ a ei a P ˆ j All components of x or p should be simultaneously measurable → xˆ , xˆ pˆ , pˆ 0 T ˆ a T ˆ b T ˆ a b e i a P e i b P e i a b P ˆ → Consider A xˆ xˆ ˆ ˆ Aˆ A xˆ → ˆ xˆ , pˆ i a Hˆ , Pˆ 0 ˆ ˆ xˆ a ei a P xˆ ei a P xˆ → A xˆ a ei a P A xˆ e i a P → i Iˆ i a Pˆ ˆ xˆ Iˆ i a Pˆ a pˆ xˆ xˆ a pˆ See Ex.5.3 for higher order terms H independent of x (translational invariant) Classical mechanics: Conserved quantity ~ L invariant under corresponding symmetry transformation Quantum mechanics: Conserved operator = generator of symmetry transformation J i i j k x j pk Example: Angular Momentum J ˆ xˆ Pˆ 1 xˆ pˆ Jˆi , Jˆ j i i j k Jˆk 1 ˆ J i J i , J j P 5.4. Quantization of a Classical System Canonical quantization scheme: Classical Quantum L qi , qi H qˆi , pˆ i pi H qi i L qi L L qi qˆi , pˆ j i i j Schrodinger qˆi t , pˆ j t i i j Heisenberg Difficulties: • Generalized coordinates may not work. Remedy: Stick with Cartesian coordinates. • Ambiguity. ˆ ˆ E.g., AB when A , B 0 • Constrainted or EM systems with Possible remedy: use L 0 qk Reminder: velocity is ill-defined in QM. 1 ˆ ˆ ˆˆ AB BA 2 Remedy: use pi . Wave functions: x x x xˆ x x x a exp a x x T → † x pˆ a i x-representation (Taylor series) i x exp a pˆ x i x x A xˆ , pˆ A xˆ , x i x A xˆ , pˆ t A xˆ , x, t i A xˆ , pˆ d 3 x x Many bodies: x1 , x2 , 3 * x A xˆ , pˆ d x x A xˆ , x i x1 , x2 , 5.5. An Example: the One-Dimensional Harmonic Oscillator L 1 1 mx 2 m 2 x 2 2 2 H xˆ , pˆ i 1 2 1 p m 2 x 2 2m 2 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: n̂ n n n Lowering (annihilation) operator: aˆ n cn n 1 Raising (creation) operator: aˆ † n bn n 1 Canonical quantization: aˆ , aˆ † 1 n n nn n aˆ †aˆ n cn n aˆ † n 1 cn cn* n 1 n 1 cn 2 → cn bn1 n n cn bn 1 bn n aˆ aˆ † n bn n aˆ n 1 bn bn* n 1 n 1 2 → bn cn1 n n bn cn 1 n aˆ , aˆ † n bn cn 2 aˆ 0 0 → 2 aˆ , nˆ aˆ , aˆ †aˆ bn* cn 1 cn 1 cn 1 2 → 2 c2 2 2 cn cn* n n n Setting cn and bn real gives a, bc abc bca 1 c1 1 c0 0 aˆ †aˆ n cn bn1 n 2 cn* bn 1 … → aˆ n n n 1 cn n 2 bn n 1 2 nˆ aˆ † aˆ aˆ † n n 1 n 1 abc bac bac bca a, b c b a, c aˆ , aˆ † aˆ aˆ † aˆ , aˆ â aˆ † , nˆ aˆ † , aˆ † aˆ aˆ † , aˆ † aˆ aˆ † aˆ † , aˆ â † n 1 † n ˆ a 0 n! Exercise: Show that if n is restricted to the values 0 and 1, then the commutator relation aˆ , aˆ † 1 must be replaced by the anti-commutator relation ˆˆ aˆ , aˆ aa † † aˆ † aˆ 1 For the harmonic oscillator, if we set aˆ m i ˆ x 2 m pˆ aˆ † m i ˆ ˆ x p 2 m then xˆ , pˆ i 1 1 Hˆ aˆ † aˆ nˆ 2 2 aˆ , aˆ † 1 → Hence, basis { | n } is also basis of the E-representation. 1 Hˆ n n n n n 2 1 n n 2 The nth excited state contains n vibrons, each of energy ω. aˆ , nˆ aˆ aˆ † , nˆ aˆ † → aˆ , Hˆ aˆ → aˆ † , Hˆ aˆ † n t e i n t n x-representation: 1 † n ˆ n a 0 n! a 0 0 where → n x → m 2 m 2 0 x N 0 exp x 2 Using x with 1 m n! 2 n /2 n /2 n d x 0 x m d x d x 0 x 0 m d x N0 m 1/4 so that dx 0 x d 1 d 1 exp x 2 exp x 2 d x 2 d x 2 n one can show where m 2 d m 2 n x N n exp x exp x 2 d x 2m Nn n! m 1/2 and dx n x 2 1 2 1 The x- and p- representations are Fourier transforms of each other. x dp p p p dx x x 1 x 2 p dp p e 1 2 i dx x e n p dp p p n where px where px x 1 p 2 Similarly i x x x x so that p p p dp p n p n p p n dx p x x n 1 2 dx e i n x px n x x n In practice, ψ (x) is easier to obtain by solving the Schrodinger eq. 2 2 V x 2m n x n n x with appropriate B.C.s i e px 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.