Published on: **Mar 4, 2016**

- 1. Topics in Quantum Theory Simon Hall 14 December 2011 Simon Hall Topics in Quantum Theory
- 2. Physical Background Motivations: Problems with classical theory: ultra-violet catastrophe of Raleigh–Jeans law concerning black-body radiation Photoelectric eﬀect: emission of electrons from metal surfaces depends on frequency (ν) of incident light; Einstein—quantization of light into packets (quanta) with energy integer multiples of E = hν = ω (where = h/2π) Double slit experiment: individual photons produce a cumulative interference pattern on photographic screen even when emitted sequentially over time Simon Hall Topics in Quantum Theory
- 3. Classical Dynamical Systems Newtownian approach: Unique coordinates in a conﬁguration space with associated masses (point masses) or, collections of coordinates whose relative positions do not change (rigid body dynamics) Underlying coordinate space is complete and ﬁnite dimensional (usually R3) The state of the system is given by each particle’s position r(t) and velocity ˙r(t). Momentum and kinetic energy are quantites derived analytically from the state vector Conservation laws are based on empirical experience Simon Hall Topics in Quantum Theory
- 4. Classical Dynamical Systems Newtownian approach (cont): The system evolves according to Newton’s laws of motion Needs extension to relativistic mechanics when the velocities in the system are not c; treats the concepts of space, time and force diﬀerently Need quantum mechanics to deal with systems with length scales on the order of their de Broglie wavelength λdB = h/p or less Simon Hall Topics in Quantum Theory
- 5. Classical Dynamical Systems Hamiltonian formalism State of a particle at time t described by the variables x(t), p(t): generalized coordinates and momenta in a two-dimensional phase space. So every dynamical variable is some function of x and p In classical theory, taking a measurement is not considered to inﬂuence the result, given “ideal” experimental conditions The state variables evolve according to Hamilton’s equations: ˙x = ∂H ∂p , ˙p = − ∂H ∂x where H = H(x, p, t) is the Hamiltonian function, identiﬁed with the total energy of the system. Simon Hall Topics in Quantum Theory
- 6. Dirac Notation & Vector Space Theory Let V be an n-dimensional abstract vector space over the ﬁeld of complex numbers C. We denote an element of V by the ket |v . Suppose we have a linearly-independent basis for V , {|1 , |2 , . . . , |n } then |v = n i=1 ai |i where the ai are complex-valued scalars. Simon Hall Topics in Quantum Theory
- 7. Dirac Notation & Vector Space Theory We deﬁne the bra v| corresponding to the ket |v as the conjugate transpose (i.e. adjoint) of |v . So if |v = n i=1 ai ei = n i=1 ai |i then v| = n i=1 ai (ei )T = n i=1 i| ai We will make use of this situation extended to inﬁnitely many dimensions: V = L2[X] where X is a subspace of R3. Simon Hall Topics in Quantum Theory
- 8. Dirac Notation & Vector Space Theory Let f , g ∈ L2[R]. Then f |g = R f (x)g(x) dx Simon Hall Topics in Quantum Theory
- 9. Dirac Notation & Vector Space Theory Let f , g ∈ L2[R]. Then f |g = R f (x)g(x) dx The inner product gives rise to a norm deﬁned by ||f || := f |f Simon Hall Topics in Quantum Theory
- 10. Linear Operators An operator Ω transforms a vector |f into another, |f : Ω |f = f or f | Ω = f A linear operator has the properties Ω(a |f + b |g ) ≡ aΩ |f + bΩ |g ( f | a + g| b)Ω ≡ f | aΩ + g| bΩ A Hermitian (or self-adjoint) operator has Ω† |f = Ω |f for all |f ∈ V where † denotes the conjugate transpose. Simon Hall Topics in Quantum Theory
- 11. The Wavefunction Experimental results led to theorists representing particles by a function or statevector |ψ , instead of coordinates and velocities. This takes the form |ψ(r, t) = ei(k·r−ωt) = cos(k · r − ωt) − i sin(k · r − ωt) (r = (x, y, z), k = (kx , ky , kz)) Simon Hall Topics in Quantum Theory
- 12. The Wavefunction Once we have a wavefunction ψ(r, t), usual interpretation: |ψ(r, t0)|2 dr is the probability of ﬁnding the particle with w.f. ψ in volume element dr at time t0—if we make a measurement. We require R3 |ψ|2 dr = R3 ψψ dr = ||ψ||2 = 1 like any self-respecting probability distribution function. Simon Hall Topics in Quantum Theory
- 13. Quantum Mechanical Operators The classical observable quantities position x and momentum p are replaced by operators ˆx → x ˆp → −i which act on the wavefunction. The expected value of an observable represented by an operator Ω of a particle with w.f. ψ is given by E[Ωψ] = ψ| Ω |ψ = R3 ψΩψ dr Simon Hall Topics in Quantum Theory
- 14. The Schrödinger Equation The quantum Hamiltonian operator is given by ˆH = ˆp · ˆp 2m + V (x) = − 2m 2 + V (x) We want to identify the Hamiltonian of the single-partcle system with its total energy, that is ˆH |ψ = E |ψ : − 2 2m 2 |ψ + V (x) = E |ψ Simon Hall Topics in Quantum Theory
- 15. The Schrödinger Equation Recall E = ω due to Einsten. Given |ψ(r, t) = ei(k·r−ωt), we ﬁnd ∂ ∂t |ψ = −iω |ψ so i ∂ ∂t |ψ = ω |ψ = E |ψ Finally, ˆH |ψ = E |ψ − 2m 2 |ψ(r, t) + V (r) = i ∂ ∂t |ψ(r, t) Quantum mechanical "equation of motion"—describes how the state of the system evolves in time. Simon Hall Topics in Quantum Theory