Advanced Quantum Physics Lecture Notes PDF

Advanced quantum physics Theoretische Physik (Quantenmechanik II) Prof. Dr. Jan Sperling Theoretical Quantum Science Institute for Photonic Quantum Systems & Department of Physics Faculty of Science date...

Advanced quantum physics Theoretische Physik (Quantenmechanik II) Prof. Dr. Jan Sperling Theoretical Quantum Science Institute for Photonic Quantum Systems & Department of Physics Faculty of Science dated: January 28, 2025 Paderborn University Contents 0. Preliminaries 4 0.1. Course evaluation................................. 5 1. Introduction 6 1.1. What is quantum physics?............................ 6 1.2. Recapitulation of basic concepts........................ 6 1.3. Description in Dirac notation.......................... 8 1.4. Archetypal applications............................. 9 1.5. Outline...................................... 10 2. Axiomatic formulation 12 2.1. Dirac–von Neumann axioms........................... 12 2.2. Hilbert spaces for quantum states........................ 13 2.3. Composite systems................................ 14 2.4. Quantum measurements............................. 15 2.5. Equations of motion............................... 18 2.6. Summary..................................... 19 3. Operator algebra 20 3.1. General concepts................................. 20 3.2. Heisenberg–Weyl algebra............................. 22 3.3. Angular momentum algebra........................... 23 3.4. Summary..................................... 25 4. Perturbation theory 26 4.1. Time-independent perturbation theory..................... 26 4.2. Dyson expansion................................. 28 4.3. Interaction picture................................ 30 4.4. Time-dependent perturbation theory...................... 30 4.5. Summary..................................... 31 5. Open quantum systems 32 5.1. Statistical operator................................ 32 5.2. Nakajima–Zwanzig equation........................... 34 5.3. Lindblad equation................................ 36 5.4. Quantum trajectories and quantum jumps................... 39 5.5. Summary..................................... 41 6. Many-body quantum systems and indistinguishable particles 42 6.1. Many-body states and entanglement...................... 42 6.2. Product approximations............................. 44 6.3. Indistinguishable quantum particles...................... 47 6.4. Summary..................................... 50 2  Contents 7. Second quantization 51 7.1. The Fock space.................................. 51 7.2. Annihilation and creation operators...................... 55 7.3. Field operators.................................. 58 7.4. Example: free Schrödinger field in second quantization............ 60 7.5. Commutator relations, Fock-operator expansions, and interactions..... 60 7.6. Summary..................................... 62 8. Dirac theory of spin-1/2 particles 64 8.1. Elements of special relativity: a recap..................... 64 8.2. Relativistic waves and Klein–Gordon equation................. 66 8.3. Derivation of the Dirac equation........................ 67 8.4. Solutions of the Dirac equation......................... 69 8.5. Transformation properties of the Dirac equation............... 71 8.6. Summary..................................... 73 A. Summary for exam prep 74 A.1. Some key concepts................................ 74 A.1.1. Fundamentals............................... 74 A.1.2. Dynamics................................. 74 A.2. Many-body systems............................... 75 A.2.1. Entanglement theory........................... 75 A.2.2. Indistingushable particles........................ 75 A.2.3. Second quantization........................... 76 3 0. Preliminaries MSc Physics course (PAUL reference: L.128.27120, teaching language: German) covered topics – axiomatic approach – operator algebra – perturbation theory – many-body systems – open quantum systems – second quantization – relativistic quantum mechanics – including examples and applications e.g., interactions, electromagnetic fields, propagators 4h lecture & 2h exercise/tutorial weekday time lecture hall lecturer lecture Tuesdays 09:15 – 10:45 A3 Jan Sperling lecture Thursdays 11:00 – 12:30 A2 Jan Sperling exercise Fridays 14:00 – 16:00 A2 Yannick Freitag schedule WS24/25 lecture dates (Tuesday) lecture dates (Thursday) exercise dates (Friday) 08. Oct. 2024 10. Oct. 2024 11. Oct. 2024 15. Oct. 2024 17. Oct. 2024 18. Oct. 2024 22. Oct. 2024 24. Oct. 2024 25. Oct. 2024 29. Oct. 2024 31. Oct. 2024 — 05. Nov. 2024 07. Nov. 2024 08. Nov. 2024 12. Nov. 2024 14. Nov. 2024 15. Nov. 2024 19. Nov. 2024 21. Nov. 2024 22. Nov. 2024 26. Nov. 2024 28. Nov. 2024 29. Nov. 2024 03. Dec. 2024 05. Dec. 2024 06. Dec. 2024 10. Dec. 2024 12. Dec. 2024 13. Dec. 2024 17. Dec. 2024 19. Dec. 2024 20. Dec. 2024 — — — 07. Jan. 2025 09. Jan. 2025 10. Jan. 2025 14. Jan. 2025 16. Jan. 2025 17. Jan. 2025 21. Jan. 2025 23. Jan. 2025 24. Jan. 2025 28. Jan. 2025 30. Jan. 2025 31. Jan. 2025 exam: 2h–3h, written........................... dates: 2025-02-11 and 2025-03-17 contact: [email protected] 4  0. Preliminaries Table 1.: Frequently used notations and symbols. quantity description H Hilbert space ∗, † complex conjugate, Hermitian conjugate |ψ⟩, ⟨ϕ| p state vector (ket), dual (bra) vector ⟨ϕ|ψ⟩, ∥ψ∥ = ⟨ψ|ψ⟩ inner product, norm L̂, 1̂ linear operator, identity Ĥ, Û Hamiltonian, unitary time-evolution operator t, ω, E time, frequency, energy ⃗ ⃗x, k, p⃗ position, wave vector, momentum (mostly in R3 ) ⃗ ∆ ∇, nabla (del in R3 ), Laplacian ·, × dot product, cross product (in R3 ) [ · , · ], { · , · } commutator, anti-commutator â, ↠, n̂ annihilation operator, creation operator, number operator ⃗ˆ ⃗ ⃗ J, L̂, Ŝ angular momentum, orbital angular momentum, spin m, e mass, electric charge (mostly elementary charge) ⃗ Φ A, magnetic vector potential, electric scalar potential ⟨L̂⟩, ∆L̂ = L̂ − ⟨L̂⟩1̂ expected value, “centralized” operator Table 2.: Few fundamental physical constants. quantity symbol (approx.) value Planck constant h = 2πℏ 6.626 070 15 × 10−34 Kg m2 s−1 speed of light in vacuum c 299 279 458 m s−1 vacuum electric permittivity ε0 8.854... × 10−12 C2 Kg−1 m−3 s2 vacuum magnetic permeability µ0 (c2 ε0 )−1 elementary charge e 1.602 176 634 × 10−19 C Boltzmann constant kB 1.380 649 × 10−23 Kg m2 s−2 K−1 0.1. Course evaluation your opininion counts! please contribute to the assessment of the lecture and exercise/tutorial lecture evaluation http://www.physik.upb.de/vkrit TAN: 8TUT9 5 1. Introduction 1.1. What is quantum physics? foremost: fundamental framework (scientific language) to speak about nature, i.e., modern way for understanding physical systems describes, supersedes, and unifies classical mechanics and classical wave theories, as well as statistical physics foundation for many fields, such as – particle physics, – condensed matter physics, – quantum field theories (incl. quantum optics), – chemistry, – etc. drives technological and scientific advancements, e.g., – lasers, diodes, and transistors in the so-called first quantum revolution – and quantum information science and technology in the (current) second quantum revolution compatible with special relativity, but currently incompatible with general relativity comes in two formulations — first and second quantization quantum mechanics is mostly associated, but definitively not limited to microscopic systems (small length scales, low energy scales, few particles, etc.) 1.2. Recapitulation of basic concepts focus on single-particle wave-function description complex-valued wave function ψ(t, ⃗x) (spatial) probability density |ψ(t, ⃗x)|2 at time t normalization 1 = d3 ⃗x |ψ(t, ⃗x)|2 , R implying a square-integrable function ψ for each time t relation to plane-wave expansion (momentum representation via ψ̃) d3 p⃗ Z   i ψ(t, ⃗x) = √ 3 exp ℏ p⃗ · ⃗x ψ̃(t, p⃗), (1.1) 2πℏ including reduced Planck constant ℏ 6  1. Introduction remark. analogously, Fourier transform ω → t for expansion in monochromatic waves possible de Broglie relation and momentum operator ℏ⃗ p⃗ = ℏ⃗k and ∇ψ = p⃗ˆψ, (1.2) i identifying momentum p⃗ and wave vector ⃗k Planck–Einstein relation and Schrödinger equation ∂ψ E = ℏω and iℏ = Ĥψ, (1.3) ∂t including Hamiltonian Ĥ separation of time variable, ψ(t, ⃗x) = exp(−iEt/ℏ)ϕ(⃗x) → time-independent Schrödinger equation, Ĥϕ = Eϕ, (1.4) i.e., eigenvalue equation of the Hamiltonian commutation relation (k, l ∈ {1, 2, 3})  0 for k ̸= l, [x̂k , p̂l ] = x̂k p̂l − p̂l x̂k = (1.5) iℏ1̂ for k = l, with identity 1̂ remark. multiples of identity often written as via the scalar without the identity, i.e., λ1̂ψ = λψ expected values, e.g., ⟨p̂l ⟩ = d3 ⃗x ψ ∗ (p̂l ψ) R variances, e.g., ⟨(∆x̂l )2 ⟩ = ⟨x̂2l ⟩ − ⟨x̂l ⟩2 , (1.6) where ∆x̂l = x̂l − ⟨x̂l ⟩1̂ uncertainty relation (Heisenberg’s indeterminacy principle)  2 2 ℏ 2 ⟨(∆x̂l ) ⟩⟨(∆p̂l ) ⟩ ≥ (1.7) 2 typical non-relativistic Hamiltonian (mass m) p⃗ˆ · p⃗ˆ ˆ), Ĥ = + V (⃗x (1.8) 2m with a potential V , and the following for electromagnetic interaction (charge e) 1 ˆ   ⃗ · p⃗ˆ − eA  ⃗ + eΦ, Ĥ = p⃗ − eA (1.9) 2m ⃗ and Φ denote the magnetic and electric potentials, respectively where A 7  1. Introduction example. one-dimensional, free particle p̂2 ℏ2 ∂ 2 – Hamiltonian Ĥ = =− 2m 2m ∂x2 – monochromatic plane-wave ansatz yields Z dk ψ(t, x) = √ eikx e−iω(k)t ψ̃0 (k) (1.10) 2π | {z } =ψ̃(t,k) with initial values ψ̃0 (k) at t = 0, where dk |ψ0 (k)|2 = 1, and dispersion R relation ℏk 2 ω(k) = (1.11) 2m – expected values (nth moment) Z ⟨p̂n ⟩ = ℏn dk |ψ̃(t, k)|2 k n Z = ℏn dk |ψ̃0 (k)|2 k n = ℏn ⟨k n ⟩0 (1.12) ∂n Z n ⟨x̂ ⟩ = in dk ψ̃(t, k)∗ n ψ̃(t, k), ∂k ∂ 1 ∂ using x → i ∂k and i ∂x → k under Fourier transform – evaluating specific expressions, we find   2 2 2 ⟨p̂⟩0 ∂ ⟨k⟩20
⟨p̂⟩ = ℏ⟨k⟩0 , ⟨(∆p̂) ⟩ = ℏ ⟨k ⟩0 − , ⟨x̂⟩ = t + i , m ∂k 0 * 2 +  2 ∂ 2  2 t 2 ∂ and ⟨(∆x̂) ⟩ = 2 ⟨(∆p̂) ⟩0 + i − i (1.13) m ∂k ∂k 0 0 tℏ  + ⟨ k − ⟨k⟩0 , i∂k − ⟨i∂k ⟩0 ⟩0 , m including the anti-commutator { · , · } 1.3. Description in Dirac notation remark. in-depth details on the context of “bra-ket” notation follow later quantum states via “ket” vectors |ψ⟩ orthonormal basis B, e.g., discrete basis B = {|n⟩ : n ∈ N} and continuous basis B = {|x⟩ : x ∈ R}, obeying – (orthonormality) ⟨b|b′ ⟩ = δ(b, b′ ) for all |b⟩, |b′ ⟩, with normalization for b = b′ and orthogonality for b ̸= b′ – examples ⟨n|n′ ⟩ = δn,n′ and ⟨x|x′ ⟩ = δ(x − x′ ), (1.14) with the Kronecker symbol δn,n′ and the Dirac delta distribution δ(x − x′ ) 8  1. Introduction Z X – (completeness relation) 1̂ = |b⟩⟨b| |b⟩∈B – examples Z X 1̂ = |n⟩⟨n| and 1̂ = dx |x⟩⟨x| (1.15) n∈N R – state expansion, e.g., X Z |ψ⟩ = ψn |n⟩ and |ψ⟩ = dx ψ(x)|x⟩ (1.16) n R duals via “bra” vectors ⟨ϕ| = |ϕ⟩† , e.g.,   ϕ † |ϕ⟩ = ϕ0 |0⟩ + ϕ1 |1⟩ = 0 7→ ⟨ϕ| = ϕ∗0 ⟨0| + ϕ∗1 ⟨1| = ϕ∗0 ϕ∗1   (1.17) ϕ1 Z X inner product as ⟨ϕ|ψ⟩ = ϕ(b)∗ ψ(b) |b⟩∈B operators Z X L̂ = 1̂L̂1̂ = L(b, b′ )|b⟩⟨b′ |, with L(b, b′ ) = ⟨b|L̂|b′ ⟩ (1.18) |b⟩,|b′ ⟩∈B and expected values ⟨L̂⟩ = ⟨ψ|L̂|ψ⟩ important families of operators – (self-adjoint/Hermitian) L̂† = L̂ – (unitary) Û † Û = 1̂ Schrödinger equation d |ψ(t)⟩ = Ĥ|ψ(t)⟩ iℏ (1.19) dt and time-independent Schrödinger equation Ĥ|ϕ⟩ = E|ϕ⟩ (1.20) 1.4. Archetypal applications example. harmonic oscillator (one-dimensional) – Hamiltonian 1 2 mω 2 2 p̂ + Ĥ = x̂ (1.21) 2m 2 1 ∂2ϕ 1 2 r E mω – rescaling to ϕ=− + ξ ϕ, where ξ = x ℏω 2 ∂ξ 2 2 ℏ – eigenfunctions → Hermite functions hn (ξ), Z |n⟩ = dξ hn (ξ)|ξ⟩ (1.22) in (scaled) position basis 9  1. Introduction – eigenvalues En = ℏω(n + 1/2), i.e., Ĥ|n⟩ = En |n⟩ – for instance, take ϕ0 (x) = N exp(sx2 ) as explicit ansatz for ground state, n = 0, 2 ℏ2 2 mω 2 2 sx2 E0 N esx = − 2s + 4s2 x2 N esx +
x Ne 2m 2 equating coefficients ℏ2 −→ x0 : E0 = − s (1.23) m 1 x : 0=0 ℏ2 2 mω 2 x2 : 0= − 2s + , m 2 implying s = ± mω 2ℏ – therefore, ϕ0 (x) = N exp − 12 ξ 2 from s < 0, with normalization constant N ,
and E0 = ℏω/2 – what happens with solution for s > 0? example. hydrogen atom – Hamiltonian 1 ˆ ˆ e2 1 Ĥ = p⃗ · p⃗ − (1.24) 2m ˆ 4πε0 ⃗x – bound-state solutions (E < 0) me4 1 Ĥ|n, l, m⟩ = − |n, l, m⟩, (1.25) 2(4πε0 ) ℏ n2 2 2 with principal quantum number n ∈ N \ {0} – azimuthal quantum number l (total angular momentum), – magnetic quantum number m (angular momentum in z direction) – both angular-momentum contributions via spherical harmonic wave function, Ylm (ϑ, φ) √ – thus, separated radial (r = |⃗x| = ⃗x · ⃗x) problem ℏ2 ∂ 2 ϕ 2 ∂ϕ ℏ2 l(l + 1) e2 1   Eϕ = − + + ϕ − ϕ 2m ∂r2 r ∂r 2m r2 4πε0 r 2mEa2B ∂ 2 ϕ 2 ∂ϕ l(l + 1) 2 (1.26) 2 ϕ = − 2 − + 2 ϕ − ϕ, | ℏ{z } ∂ρ ρ ∂ρ ρ ρ =η 4πε0 ℏ2 where ρ = r/aB and the Bohr radius aB = me2 – bound solutions via generalized Laguerre polynomials 1.5. Outline in this course – axiomatic formulation of quantum physics 10  1. Introduction – operator algebra – perturbation theory – open quantum systems – many-body quantum systems – second quantization (first quantization is what you know from introduction to quantum mechanics) – relativistic quantum physics (exemplified by Dirac particle) further courses, e.g., – quantum information theory – relativistic quantum field theory 11  2. Axiomatic formulation goal: rigorous formulation of the principles of quantum physics like for all principles in physics, postulates present a mathematical context that unify a number of (all known) ob- servations remark. almost everything here should be known from previous lectures on quantum mechan- ics but is presented in a slightly different context 2.1. Dirac–von Neumann axioms axioms/postulates based on and extensions of formulations by Dirac and von Neu- mann postulates. overview axioms: (i) (State vectors.) The state of an isolated physical system is represented by a state vector |ψ⟩ belonging to a Hilbert space H. (ii) (Composite system postulate.) The Hilbert space of a composite system is the tensor product of the spaces associated with the components.1 (iii) (Observables.) A measurable physical quantity L̂—an observable—is described by a Hermitian operator acting on H.2 The result of measuring a physical quantity must be one of the eigenvalues of the observable. (iv) (Born’s rule.) When the physical quantity is measured on a system in a nor- malized state, the probability of obtaining an eigenvalue of the corresponding observable L̂ is given by the amplitude squared of the appropriate wave func- tion; that is, the norm of the projection onto corresponding eigenspace. (v) (Collapse postulate.) The state of the system immediately after the measure- ment of L̂ is the normalized projection of the initial state onto the eigensubspace associated with the measured eigenvalue.3 (vi) (Schrödinger equation.) The time evolution of a state vector is governed by the Schrödinger equation, which includes the Hamiltonian that is an observable associated with the total energy of the system.4 (vii) (Symmetrization postulate.) The wavefunction of a system of identical particles is either totally symmetric (Bosons) or totally antisymmetric (Fermions) under interchange of any pair of particles. in the following, discuss several postulates 1 For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles. 2 The eigenvectors of L̂ form a basis for H. 3 Equivalently, we have continuity of measurement outcomes. 4 Likewise, we have a unitary propagation in time via a first-order differential equation. 12  2. Axiomatic formulation others will be discussed in separate chapters, and we revisit postulates in later chapters also furthermore, extensions and generalizations will be considered in the continuation of this course 2.2. Hilbert spaces for quantum states consider a single quantum particle postulate (i) → state of the system is given by an element of a Hilbert space H, which is a complex and complete inner product space, complex vector (i.e., linear) space, |ψ + ϕ⟩ = |ψ⟩ + |ϕ⟩ ∈ H (additive) (2.1) and |λψ⟩ = λ|ψ⟩ ∈ H (homogeneous) for all |ψ⟩ ∈ H, |ϕ⟩ ∈ H (“ket vectors”), and λ ∈ C plus the common rules for addition of vectors and multiplication with scalars, and combination thereof (distributive laws) inner product ⟨ϕ|ψ⟩, with ⟨ϕ|λψ⟩ = λ⟨ϕ|ψ⟩, ⟨λϕ|ψ⟩ = λ∗ ⟨ϕ|ψ⟩, ⟨ϕ|ψ + ψ ′ ⟩ = ⟨ϕ|ψ⟩ + ⟨ϕ|ψ ′ ⟩, (2.2) ′ ′ ⟨ϕ + ϕ |ψ⟩ = ⟨ϕ|ψ⟩ + ⟨ϕ |ψ⟩, and ⟨ϕ|ψ⟩∗ = ⟨ψ|ϕ⟩ (symmetry), inner product induces a norm  p ⟨ψ|ψ⟩ = 0 if |ψ⟩ = 0, ∥ψ∥ = ⟨ψ|ψ⟩, with (2.3) ⟨ψ|ψ⟩ > 0 otherwise remarks. a few comments on notation – H can have infinite dimensionality, but is separable in physics; separable Hilbert space: countable, orthonormal basis exists5 – a complete space is one in which every Cauchy sequence converges with respect to the norm under study homogeneous → quantum states which are distinct just by a non-zero factor are considered to describe the same state – i.e., equate |ψ⟩ = λ|ψ ′ ⟩ for λ = |λ|ei arg(λ) ∈ C \ {0} – without loss of generality, ∥ψ∥ = 1 via |λ| = 1/∥ψ ′ ∥ as choice of normalization, – ignore global phases since arg(λ) free parameter 5 but not all bases have to be countable 13  2. Axiomatic formulation dual vectors ⟨ϕ| (“bra vectors”) via Riesz representation theorem: for each f : H → C linear (bounded) function exists one ⟨ϕ| such that f (|ψ⟩) = ⟨ϕ|ψ⟩ important inequality: Cauchy–Schwarz inequality (CSI) |⟨ϕ|ψ⟩| ≤ ∥ϕ∥ ∥ψ∥ (2.4) proof. CSI follows from the positivity of the norm – for |ψ⟩, |ϕ⟩ ∈ H \ {0} and λ ∈ C, consider and minimize for all λ nonnegative z }| {   0 ≤ ⟨ψ + λϕ|ψ + λϕ⟩ = ⟨ψ|ψ⟩ + |λ|2 ⟨ϕ|ϕ⟩ + 2|λ| Re ei arg λ ⟨ψ|ϕ⟩ (2.5) | {z } ≥−|⟨ψ|ϕ⟩|, e.g., arg λ=π−arg⟨ψ|ϕ⟩ – the right-hand side, e.g., via the derivative with respect to |λ|, is further mini- mized by |⟨ψ|ϕ⟩| |⟨ψ|ϕ⟩|2 |⟨ψ|ϕ⟩|2 |λ| = ⇒ 0 ≤ ⟨ψ|ψ⟩ + −2 (2.6) ⟨ϕ|ϕ⟩ ⟨ϕ|ϕ⟩ ⟨ϕ|ϕ⟩ – rewriting this equations proves the CSI in the form |⟨ψ|ϕ⟩|2 ≤ ⟨ψ|ψ⟩⟨ϕ|ϕ⟩ 2.3. Composite systems postulate (ii) → independent degrees of freedom via tensor-product Hilbert spaces focus on two degrees of freedom for the time being consider HA and HB in this manner, together with states |ψA ⟩ ∈ HA and |ψB ⟩ ∈ HB , resulting in the joint state |ψA:B ⟩ = |ψA ⟩ ⊗ |ψB ⟩, (2.7) a simple tensor, or tensor-product state from HAB = HA ⊗ HB recall. property of being linear in each component (i.e., bilinear for two) (λ|ψA ⟩ + κ|ϕA ⟩) ⊗ |ψB ⟩ = λ|ψA ⟩ ⊗ |ψB ⟩ + κ|ϕA ⟩ ⊗ |ψB ⟩, (2.8) |ψA ⟩ ⊗ (λ|ψB ⟩ + κ|ϕB ⟩) = λ|ψA ⟩ ⊗ |ψB ⟩ + κ|ψA ⟩ ⊗ |ϕB ⟩ since HAB vector space, linear combinations allowed as well, e.g., X |ψAB ⟩ = λj |ψA,j ⟩ ⊗ |ψB,j ⟩ (2.9) j it is important to emphasize that |ψAB ⟩ does not necessarily admit a tensor-product structure, motivating the notion of entanglement |ψAB ⟩ ∈ HAB entangled def. (2.10) ⇔ ∀|ψA ⟩ ∈ HA , |ψB ⟩ ∈ HB : |ψAB ⟩ = ̸ |ψA ⟩ ⊗ |ψB ⟩ 14  2. Axiomatic formulation entangled states are not described through states of a single state for each degree of freedom (but a quantum superposition thereof) the inner product of HA ⊗ HB induced by components’ inner products, from HB X z }| { ⟨ψAB |ϕAB ⟩ = λ∗j κj ′ ⟨ψA,j |ϕA,j ′ ⟩ ⟨ψB,j |ϕB,j ′ ⟩ (2.11) j,j ′ | {z } from HA all extends to arbitrary number of degrees of freedom, e.g., |ψ1 ⟩ ⊗ |ψ2 ⟩ ⊗ · · · ⊗ |ψN ⟩ ∈ H1 ⊗ H2 ⊗ · · · ⊗ HN (2.12) for an N -particle state furthermore, postulate (vii) → description of indistinguishable particles, not independent (i.e., dependent) degrees of freedom symmetric and anti-symmetric states expressed via symmetric and anti-symmetric tensor-products, e.g., |ψboson ⟩ = |ψA ⟩ ∨ |ψB ⟩ = |ψA ⟩ ⊗ |ψB ⟩ + |ψB ⟩ ⊗ |ψA ⟩, (2.13) |ψfermion ⟩ = |ψA ⟩ ∧ |ψB ⟩ = |ψA ⟩ ⊗ |ψB ⟩ − |ψB ⟩ ⊗ |ψA ⟩, and linear combinations thereof further discussions follow in the context of many-body systems and second quanti- zation 2.4. Quantum measurements postulates (iii), (iv), and (v) pertain to measurements → – notion of observables and measurement outcomes, – outcome probability (Born’s rule), – and the post-measurement state (collapse of the wave function) recall. concept of Hermitian conjugate L̂† : for all vectors holds6 def. ⟨L̂† ϕ|ψ⟩ = ⟨ϕ|L̂ψ⟩, (2.14) can also write ⟨ϕ|L̂|ψ⟩∗ = ⟨ψ|L̂† |ϕ⟩, (2.15) where L̂ and L̂† act on vector to the right notion observable: self-adjoint/Hermitian operator, L̂ = L̂† spectral decomposition (spectral theorem): XZ L̂ = λ(b) |b⟩⟨b| (2.16) |b⟩∈B for an orthonormal basis B of eigenvectors and real-valued eigenvalues λ(b), i.e., L̂|b⟩ = λ(b)|b⟩ (2.17) 6 a dense subset of all vectors suffices 15  2. Axiomatic formulation spectrum: set of all eigenvalues σ(L̂) = {λ(b) : |b⟩ ∈ B} all statesZfor one eigenvalue via projection-valued measure X Π̂(λ) = δ(λ(b), λ)|b⟩⟨b| |b⟩∈B rewrite spectral decomposition Z Z Z ! X L̂ = dλ λ Π̂(λ) = dλ δ(λ(b), λ)λ |b⟩⟨b| (2.18) λ∈σ(L̂) |b⟩∈B λ∈σ(L̂) Z note. relations Π̂(λ)Π̂(λ′ ) = δ(λ, λ′ )Π̂(λ) and dλ Π̂(λ) = 1̂ λ∈σ(L̂) measurement outcomes: λ ∈ σ(L̂) instead of a single measurement {λ}, multiple measurements Λ possible measured probability: Z p(λ ∈ Λ) = dλ ⟨ψ|Π̂(λ)|ψ⟩ , (2.19) λ∈Λ | {z } probability mass function or probability density with Λ ⊂ σ(L̂) state after measurement (collapsed state |ψ ′ ⟩) – measurement yields λ with probability p(λ) – suppose second measurement of L̂ directly after the first measurement – not time to evolve → second outcome λ′ = λ with certainty i.e., “continuity of measurement outcomes” no random jumps of measurement results with zero time evolution – likewise, conditional probability ⟨ψ ′ |Π̂(λ′ )|ψ ′ ⟩ = δ(λ′ , λ) – thus, state |ψ ′ ⟩ after measurement in the eigenspace spanned by Π̂(λ), and properly normalized Π̂(λ)|ψ⟩ |ψ ′ ⟩ = (2.20) ∥Π̂(λ)ψ∥ mean value Z Z ⟨L̂⟩ = dλ ⟨ψ|Π̂(λ)|ψ⟩λ = ⟨ψ| dλ λ Π̂(λ)|ψ⟩ = ⟨ψ|L̂|ψ⟩ = tr(L̂|ψ⟩⟨ψ|) σ(L̂) σ(L̂) (2.21) variance via ∆L̂ = L̂ − ⟨L̂⟩1̂ ⟨(∆L̂)2 ⟩ = ⟨L̂⟩2 − 2⟨L̂⟩⟨L̂⟩ + ⟨L̂⟩2 ⟨1̂⟩ = ⟨L̂2 ⟩ − ⟨L̂⟩2 (2.22) |{z} =1 observables L̂ and M̂ 16  2. Axiomatic formulation – commutator [L̂, M̂ ] = 0 → joint spectral decomposition, measurable at same time – commutator [L̂, M̂ ] ̸= 0, i.e., L̂M̂ ̸= M̂ L̂ → no joint spectral decomposition, different collapses p(λ|µ) ̸= p(µ|λ) ̸= δ(µ, λ) (no certainty) proof. uncertainty relation – Cauchy–Schwarz inequality, LHS = ⟨α|α⟩⟨β|β⟩ ≥ |⟨α|β⟩|2 = RHS – let  = † and B̂ = B̂ † be two observables, as well as a, b ∈ R – let |α⟩ = ( − a1̂)|ψ⟩ and |α⟩ = (B̂ − b1̂)|ψ⟩ – thus, LHS = ⟨ψ|( − a1̂)† ( − a1̂)|ψ⟩⟨ψ|(B̂ − b1̂)† (B̂ − b1̂)|ψ⟩ (2.23) = ⟨ψ|( − a1̂)2 |ψ⟩⟨ψ|(B̂ − b1̂)2 |ψ⟩ and 2 RHS = ⟨ψ|( − a1̂)† (B̂ − b1̂)|ψ⟩  2   2 = Re ⟨ψ|( − a1̂)(B̂ − b1̂)|ψ⟩ + Im ⟨ψ|( − a1̂)(B̂ − b1̂)|ψ⟩ 1 2 = ⟨ψ|( − a1̂)(B̂ − b1̂)|ψ⟩ + ⟨ψ|(B̂ − b1̂)( − a1̂)|ψ⟩ 2 2 1   + ⟨ψ|( − a1̂)(B̂ − b1̂)|ψ⟩ − ⟨ψ|(B̂ − b1̂)( − a1̂)|ψ⟩ 2i 1 2 1 2 = ⟨ψ|{ − a1̂, B̂ − b1̂}|ψ⟩ + ⟨ψ|[ − a1̂, B̂ − b1̂]|ψ⟩ 4 4 1 2 ≥ ⟨ψ|[Â, B̂]|ψ⟩ 4 (2.24) – set a = ⟨ψ|Â|ψ⟩ and b = ⟨ψ|B̂|ψ⟩, yielding 1 2 1 2 ⟨ψ|(∆Â)2 |ψ⟩⟨ψ|(∆B̂)2 |ψ⟩ ≥ ⟨ψ|{∆Â, ∆B̂}|ψ⟩ + ⟨ψ|[Â, B̂]|ψ⟩ (2.25) | 4 {z 4 } Robertson–Schrödinger uncertainty relation and 1 2 ⟨ψ|(∆Â)2 |ψ⟩⟨ψ|(∆B̂)2 |ψ⟩ ≥ ⟨ψ|[Â, B̂]|ψ⟩ (2.26) | {z 4 } Heisenberg uncertainty relation remark.. quantity ⟨ 12 {∆L̂, ∆M̂ }⟩ relates to symmetric covariance, 1 ⟨L̂M̂ ⟩ + ⟨M̂ L̂⟩ ⟨ {∆L̂, ∆M̂ }⟩ = − ⟨L̂⟩⟨M̂ ⟩, (2.27) 2 2 which simplifies to common expression ⟨(∆L̂)(∆M̂ )⟩ = ⟨L̂M̂ ⟩−⟨L̂⟩⟨M̂ ⟩ if observables commute, L̂M̂ = M̂ L̂ 17  2. Axiomatic formulation 2.5. Equations of motion the postulate last not-yet discussed postulate (vi) → dynamics of the system distinct from measurement-induced changes of states (collapse), as discussed above Schrödinger equation d iℏ |ψ(t)⟩ = Ĥ|ψ(t)⟩, (2.28) dt with initial value |ψ(t0 )⟩ linear, homogeneous, first-order differential equation superposition principle applies to solutions and a linear time-evolution operator that is independent of the state exists with |ψ(t)⟩ = Û (t, t0 )|ψ(t0 )⟩ (2.29) assume time-independent Hamiltonian time-evolution operator ∞ 1 t − t0 m m     t − t0 X Û (t, t0 ) = exp Ĥ = Ĥ , (2.30) iℏ m! iℏ m=0 def. d where Ĥ 0 = 1̂ and with Û (t0 , t0 ) = 1̂ and dt Û (t, t0 ) = Ĥ Û (t, t0 ) for satisfying the Schrödinger equation for projector, equation of motion as von Neumann equation follows from the product rule     d 1 1 (|ψ(t)⟩⟨ψ(t)|) = Ĥ|ψ(t) ⟨ψ(t)| + |ψ(t)⟩ − ⟨ψ(t)|Ĥ dt iℏ iℏ (2.31) 1 = [Ĥ, |ψ(t)⟩⟨ψ(t)], iℏ where Ĥ = Ĥ † was used time-dependent expectation values ∂ L̂ – suppose observable L̂, potentially time-dependent ∂t ̸= 0 – time-dependent expectation value     ⟨L̂⟩ = tr L̂(t)|ψ(t)⟩⟨ψ(t)| = tr L̂(t)Û (t, t0 )|ψ(t0 )⟩⟨ψ(t0 )|Û (t, t0 )†   (2.32) = tr Û (t, t0 )† L̂(t)Û (t, t0 )|ψ(t0 )⟩⟨ψ(t0 ) – resulting states and operators in Heisenberg picture |ψH ⟩ = |ψ(t0 )⟩ and L̂H (t) = Û (t, t0 )† L̂(t)Û (t, t0 ), (2.33) with initial (i.e., time-independent) states only 18  2. Axiomatic formulation – Heisenberg equation of motion from time-evolution operator  † d 1 ∂ L̂(t) L̂H (t) = Ĥ Û (t, t0 ) L̂(t)Û (t, t0 ) + Û (t, t0 )† Û (t, t0 ) dt iℏ ∂t   1 + Û (t, t0 )† L̂(t) Ĥ Û (t, t0 ) (2.34) iℏ ! 1 ∂ L̂(t) = [L̂H (t), Ĥ] + , iℏ ∂t H using that a time-independent Ĥ commutes with the time-evolution operator further, time-independent Schrödinger equation as eigenvalue problem, i.e., separation of variables |ψ(t)⟩ = T (t)|ϕ⟩ yields dT (t) iℏ |ϕ⟩ = T (t)Ĥ|ϕ⟩ dt (2.35) dT (t) → iℏ = ET (t) and Ĥ|ϕ⟩ = E|ϕ⟩ dt the constant of separation E allows one to solve the time-dependent part → T (t) = T (t0 ) exp(−i[t − t0 ]E/ℏ) time-independent part via eigenvalue problem/spectral decomposition, with eigen- values E = E(b) and orthonormal eigenvectors |ϕ⟩ = |b⟩ ∈ B, linearity results in solution of form Z X |ψ(t)⟩ = e−i[t−t0 ]E(b)/ℏ ψb (t0 )|b⟩ (2.36) |b⟩∈B 2.6. Summary discussion of basic axioms of quantum physics pertaining to – quantum states in Hilbert space – the description of composite systems – theory of measurements – equations of motion following this, more on operators as they are central to the language of quantum physics 19  3. Operator algebra operators as key ingredient of quantum physics previously mentioned – observables via Hermitian operators, L̂† = L̂ – unitary time-evolution operators, Û † = Û −1 here, discussion of broader concepts and specific examples of physical importance 3.1. General concepts consider linear operators L̂ : H → H concatenation of operators M̂ L̂, e.g., matrix product for finite dimensional H define commutators and anti-commutators as [L̂, M̂ ] = L̂M̂ − M̂ L̂ and {L̂, M̂ } = L̂M̂ + M̂ L̂, (3.1) e.g., used in uncertain relations, equations of motions, etc. properties and relations of commutators (proof as homework assignment, aka Lie algebra identities) – linear in both components – anti-symmetric [L̂, M̂ ] = −[M̂ , L̂], (equivalently [L̂, L̂] = 0) – product rule [ÂB̂, Ĉ] = [Â, Ĉ]B̂ + Â[B̂, Ĉ] as well as [Â, B̂ Ĉ] = [Â, B̂]Ĉ + B̂[Â, Ĉ] – Jacobi identity 0 = [Â, [B̂, Ĉ]] + [B̂, [Ĉ, Â]] + [Ĉ, [Â, B̂]] trace of an operator Z X tr(L̂) = ⟨b|L̂|b⟩ (3.2) |b⟩∈B remark. sum/integral does not necessarily exist, e.g., trace of 1̂ in infinite-dimensional H properties of the trace (homework) – linear – invariant under orthonormal basis choice B – invariant under transposition tr(L̂) = tr(L̂T ), as well as tr(L̂)∗ = tr(L̂∗ ) for complex conjugation and tr(L̂) = tr(L̂† )∗ for Hermitian conjugation1 1 aka conjugate transpose and Hermitian transpose 20  3. Operator algebra – trace of products tr(L̂M̂ ) = tr(M̂ L̂), implying cyclic property tr(ÂB̂ Ĉ) = tr(B̂ Ĉ Â) = tr(Ĉ ÂB̂) as well as tr([L̂, M̂ ]) = 02 applications – trace of dyadic product Z Z ! X X tr(|ψ⟩⟨ϕ|) = ⟨b|ψ⟩⟨ϕ|b⟩ = ⟨ϕ| |b⟩⟨b| |ψ⟩ = ⟨ϕ|ψ⟩, (3.3) |b⟩∈B |b⟩∈B | {z } =1̂ showing that tr(|ψ⟩⟨ψ|L̂) = ⟨ψ|L̂|ψ⟩ when choosing |ϕ⟩ = L̂|ψ⟩ – selecting B as the eigenbasis of L̂, then tr(L̂) is the total of eigenvalues, Z X XZ tr(L̂) = ⟨b| L̂|b⟩ = λ(b) (3.4) |b⟩∈B |{z} |b⟩∈B =λ(b)|b⟩ specifically noteworthy families of operators – unitary Û † = Û −1 – self-adjoint (Hermitian) L̂† = L̂ – positive semidefinite operators ∀|ψ⟩ ∈ H : ⟨ψ|L̂|ψ⟩ ≥ 0 (3.5) (positive definite for strictly greater than zero) one writes L̂ ≥ 0 – trace class tr(|L̂|) < ∞, (3.6) p def. with |L̂| = L̂† L̂ (|L̂| = L̂ for L̂ ≥ 0) – projections Π̂2 = Π̂ (recall spectral decomposition) – etc. functions of operators f (L̂) ∞ def. X typically via Taylor series, f (L̂) = fn L̂n , with L̂0 = 1̂ n=0   example. exponential function for time evolution, exp t−t0 iℏ Ĥ for normal operators (admitting a spectral decomposition), we have Z X
f (L̂) = f λ(b) |b⟩⟨b| (3.7) |b⟩∈B 2 if convergence is not a problem; think about [x̂, p̂] = iℏ1̂ 21  3. Operator algebra proof. conclusion from products of commuting operators – observe Z X Z X M̂ = µ(b)|b⟩⟨b|, and L̂ = λ(b′ )|b′ ⟩⟨b′ | |b⟩∈B |b′ ⟩∈B Z X ′ ′ ′ Z X (3.8) → M̂ L̂ = µ(b)λ(b )|b⟩ ⟨b|b ⟩ ⟨b | = µ(b)λ(b)|b⟩⟨b|s |b⟩,|b′ ⟩∈B | {z } |b⟩∈B =δ(b,b′ ) – apply for induction, M̂ = L̂n−1 , yielding Z n X
n L̂ = λ(b) |b⟩⟨b| (3.9) |b⟩∈B – substituting in Taylor series for f proves claim p̂2 example. free relativistic particle Ĥ = c2 p̂2 + m2 c4 ≈ mc2 + 2m p (from exercise) p2 Z p Z   2 2 2 4 2 Ĥ = dp c p + m c |p⟩⟨p| ≈ dp mc + |p⟩⟨p| (3.10) R R 2m 3.2. Heisenberg–Weyl algebra inspired by harmonic oscillators and fundamental commutator relations consider general operators x̂ and p̂, obeying the commutator relations [x̂, p̂] = ẑ, [x̂, ẑ] = 0, and [p̂, ẑ] = 0, (3.11) here specifically ẑ = iℏ1̂ introduce ladder operators, here: annihilation operator √   1 i â = √ mωx̂ + √ p̂ (3.12) 2ℏ mω and creation operator √   † 1 i â = √ mωx̂ − √ p̂ , (3.13) 2ℏ mω using x̂† = x̂ and p̂† = p̂ remark. annihilation and creation operators are non-Hermitian, â ̸= ↠new fundamental commutation relations (also resulting in Heisenberg–Weyl algebra) [â, ↠] = 1̂ (3.14) define (occupation) number operator n̂ = ↠â example. Hamiltonian for harmonic oscillator 1 2 mω 2 2   1 Ĥ = p̂ + x̂ = ℏω n̂ + (3.15) 2m 2 2 22  3. Operator algebra number operator is positive semidefinite, ⟨ψ|n̂|ψ⟩ = ∥âψ∥2 ≥ 0 known ground state n̂|0⟩ = 0 = 0|0⟩ (Gaussian wave function from previous lecture) action of creation operator on eigenstates – consider ↠1̂ = ↠[â, ↠] = n̂↠− ↠n̂ n̂↠= ↠n̂ + 1̂
→ (3.16) – acting on |n⟩ eigenvalue   z }| {   n̂ ↠|n⟩ = ↠(n + 1)|n⟩ = (n + 1) ↠|n⟩ (3.17) – thus, ↠|n⟩ = λ|n + 1⟩ – let arg(λ) = 0 and =n̂+1̂ 1 n+1 z}|{ 1 = ⟨n + 1|n + 1⟩ = 2 ⟨n| â↠|n⟩ = (3.18) |λ| |λ|2 – therefore √ ↠|n⟩ = n + 1|n + 1⟩ (3.19) analogously, annihilation operator √ â|n⟩ = n|n − 1⟩ (3.20) and â|0⟩ = 0 because |ψ⟩ = â|0⟩ obeys 0 = ⟨0|n̂|0⟩ = ∥ψ∥2 example. consider similar approach with anti-commutators instead – let |0⟩ = [ 10 ] and |1⟩ = [ 01 ], as well as     0 1 † 0 0 â = and â = (3.21) 0 0 1 0 – find {â, â} = 0 = {↠, ↠} and {â, ↠} = 1̂, as well as √ √ â|n⟩ = n|n − 1⟩ and ↠|n⟩ = 1 − n|n + 1⟩ (3.22) for n ∈ {0, 1} 3.3. Angular momentum algebra angular momentum [Jˆx , Jˆy ] = iℏJˆz , [Jˆy , Jˆz ] = iℏJˆx , and [Jˆz , Jˆx ] = iℏJˆy , (3.23) with self-adjoint Jˆw for w ∈ {x, y, z} ℏ2 ˆ 2 e.g., Heisenberg uncertainty relation: 4 |⟨Jz ⟩| ≤ ⟨(∆Jˆx )2 ⟩⟨(∆Jˆy )2 ⟩ 23  3. Operator algebra ⃗ ⃗ ⃗ decomposition into two parts, Jˆ = L̂ + Ŝ – position-dependent part: ⃗ orbital angular momentum L̂, as ⃗ L̂ = ⃗x̂ × p̂⃗ (3.24) – spin as position-independent part, [Ŝw , x̂w′ ] = 0 (3.25) for w, w′ ∈ {x, y, z} – also, both parts commute in all components, [L̂w , Ŝw′ ] = 0 total angular momentum Jˆ2 = Jˆx2 + Jˆy2 + Jˆz2 (3.26) commutes with components, [Jˆ2 , Jˆw ] = 0 define ladder operators Jˆ± = Jˆx ± iJˆy (3.27) † Hermitian conjugates to each other Jˆ± = Jˆx† ∓ iJˆy† = Jˆ∓ commutators [Jˆ2 , Jˆ± ] = 0, [Jˆ+ , Jˆ− ] = 2ℏJˆz , and [Jˆz , Jˆ± ] = ±ℏJˆ± (3.28) product   † ˆ Jˆ± J± = Jˆx2 + Jˆy2 ± i Jˆx Jˆy − Jˆy Jˆx = Jˆx2 + Jˆy2 ∓ ℏJˆz (3.29) expansion total angular momentum † ˆ 1 † ˆ 1 † ˆ Jˆ2 = Jˆ± J± ± ℏJˆz + Jˆz2 = Jˆ+ J+ + Jˆ− J− + Jˆz2 (3.30) 2 2 since Jˆ2 and, e.g., Jˆz commute, joint diagonalization possible Jˆ2 |j, m⟩ = ℏ2 j(j + 1)|j, m⟩ and Jˆz |j, m⟩ = ℏm|j, m⟩, (3.31) choosing the form j(j + 1) for convenience (with j ≥ 0) remark. by definition, Jˆ2 ≥ Jˆz2 ≥ 0 follows apply ladder operator   Jˆz Jˆ± |j, m⟩ = Jˆ± Jˆz ± ℏJˆ± |j, m⟩ = ℏ (m ± 1) Jˆ± |j, m⟩, (3.32) implying Jˆ± |j, m⟩ = λ|j, m ± 1⟩ analogously to previous section, select λ ≥ 0 with a global phase arg λ = 0; also, j unchanged as Jˆz commutes with Jˆ2 , not affecting j 24  3. Operator algebra normalization from † ˆ ⟨j, m|Jˆ2 |j, m⟩ = ⟨j, m|Jˆ± J± |j, m⟩ ±ℏ ⟨j, m|Jˆz |j, m⟩ + ⟨j, m|Jˆz2 |j, m⟩ (3.33) | {z } | {z } | {z } | {z } =ℏ2 j(j+1) =λ2 =ℏm =ℏ2 m2 therefore Jˆ± |j, m⟩ = ℏ j(j + 1) − m(m ± 1)|j, m ± 1⟩ p (3.34) special cases Jˆ+ |j, +j⟩ = 0 and Jˆ− |j, −j⟩ = 0 (3.35) thus, −j ≤ m ≤ +j with increments of one, 1 further implying j ∈ N 2 example. spin- 21 particle (ignoring orbital angular momentum) – total angular momentum s = 1/2 and spin in z direction ms = ±1/2, resulting in states states | 12 , + 12 ⟩ = |↑⟩ = [ 10 ] and | 12 , − 12 ⟩ = |↓⟩ = [ 01 ] – spin matrices Ŝw = ℏ2 σ̂w for w ∈ {x, y, z} and Pauli matrices σ̂w ,       ℏ 0 1 ℏ 0 −i ℏ 1 0 Ŝx = , Ŝy = , and Ŝz = (3.36) 2 1 0 2 i 0 2 0 −1 – total spin 3ℏ2 1 0   Ŝ 2 = , (3.37) 2 0 1 with s(s + 1) = 3/4 – ladder operators     0 1 0 0 Ŝ+ = ℏ and Ŝ− = ℏ , (3.38) 0 0 1 0 leading to Ŝ+ |↑⟩ = 0, Ŝ+ |↓⟩ = |↑⟩, Ŝ− |↓⟩ = 0, and Ŝ− |↑⟩ = |↓⟩ 3.4. Summary study of quantum-mechanical operators including commutators, as well as anti-commutators trace of operators (analytic) functions of operators prominent examples – Heisenberg–Weyl algebra – angular momentum algebra, incl. spin 25  4. Perturbation theory not all problems come with exact solution for instance, complexly interacting many-body quantum systems general approaches to approximate sought-after results → perturbation theory here, focus on first-order perturbations (methods described apply likewise for higher orders) two cases: – time-independent problems – time-dependent problems 4.1. Time-independent perturbation theory suppose Ĥ ′ = Ĥ + δ Ĥ, with perturbation δ Ĥ let Ĥ|En ⟩ = En |En ⟩ be the solutions of the unperturbed eigenvalue problem remark. here, focus on discrete and non-degenerate scenarios solutions to eigenvalue problem of perturbed Hamiltonian suppose (basis expansion) ! X |En′ ⟩ = Nn |En ⟩ + ∆n,m |Em ⟩ , (4.1) m:m̸=n | {z } =|δEn ⟩ with normalization Nn , and En′ = En + δEn substitute in eigenvalue equation   Ĥ + δ Ĥ (|En ⟩ + |δEn ⟩) = (En + δEn ) (|En ⟩ + |δEn ⟩) (4.2) Ĥ|En ⟩ + δ Ĥ|En ⟩ + Ĥ|δEn ⟩ = En |En ⟩ + δEn |En ⟩ + En |δEn ⟩, ignoring quadratic perturbations zeroth-order terms okay (unperturbed eigenvalue problem) first-order corrections   – consider projections ⟨Em | δ Ĥ|En ⟩ + Ĥ|δEn ⟩ = ⟨Em | (δEn |En ⟩ + En |δEn ⟩) – contribution for m = n ⟨En |δ Ĥ|En ⟩ + En ⟨En |δEn ⟩ = δEn + En ⟨En |δEn ⟩ (4.3) | {z } | {z } =0 =0 26  4. Perturbation theory – contribution for m ̸= n ⟨Em |δ Ĥ|En ⟩ + ⟨Em |Ĥ|δEn ⟩ = δEn ⟨Em |En ⟩ +En ⟨Em |δEn ⟩ | {z } | {z } | {z } =Em ∆n,m =0 =∆n,m (4.4) ⟨Em |δ Ĥ|En ⟩ → ∆n,m = En − Em thus, first-order perturbation yield eigenvalues and eigenvectors of the form   X ⟨Em |δ Ĥ|En ⟩ En′ = En + ⟨En |δ Ĥ|En ⟩ and |En′ ⟩ = Nn |En ⟩ + |Em ⟩ En − Em m:m̸=n (4.5) remarks. – normalization Nn :  −1/2 X |⟨Em |δ Ĥ|En ⟩|2 1 = ⟨En′ |En′ ⟩ → Nn = 1 +  (4.6) (En − Em )2 m:m̸=n – note that   X 1 |δEn ⟩ =  |Em ⟩⟨Em | δ Ĥ|En ⟩ (4.7) En − Em m:m̸=n relates to Green’s function (operator inverse) – higher-order correction via Ĥ ′ = Ĥ + δ Ĥ + δ 2 Ĥ and equating all terms of same order, e.g., δ Ĥ|δEn ⟩, δ 2 Ĥ|En ⟩, and Ĥ|δ 2 En ⟩ for the second order – consider degenerate eigenvalues via eigenspaces (span of all eigenvectors to the same eigenvalue) rather than a single eigenvector (one-dimensional eigenspace) example. let Ĥ = ℏω(n̂ + 12 1̂) and (for κ small) δ Ĥ = ℏκ(â + ↠)3 = ℏκ(â3 + 3↠â2 + 3â†3 â + â†3 + 3â + 3↠), (4.8) using commutation relations – δ Ĥ resembles is cubic potential as perturbation for quadratic Hamiltonian – we have |En ⟩ = |n⟩ and En = ℏω(n + 1/2), from Ĥ|n⟩ = ℏω(n + 1/2)|n⟩ – useful identity δ Ĥ p √ 3 |n⟩ = n(n − 1)(n − 2)|n − 3⟩ + 3 n |n − 1⟩ κℏ (4.9) √ 3 p + 3 n + 1 |n + 1⟩ + (n + 1)(n + 2)(n + 3)|n + 3⟩ – from above relations, eigenvalues En′ = En + 0 – and eigenvectors p κ n(n − 1)(n − 2) √ 3 |En′ ⟩ ∝ |n⟩ + |n − 3⟩ + 3 n |n − 1⟩ ω 3 p ! (4.10) √ 3 (n + 1)(n + 2)(n + 3) −3 n + 1 |n + 1⟩ − |n + 3⟩ 3 27  4. Perturbation theory remark. other approaches, especially Rayleigh–Ritz method, discussed in tutorial/exercise continue with time-dependent approach 4.2. Dyson expansion consider time-dependent Hamiltonian Ĥ(t) Schrödinger equation iℏ dt d |ψ(t)⟩ = Ĥ(t)|ψ(t)⟩ and |ψ(t)⟩ = Û (t, t0 )|ψ(t0 )⟩ equation of motion for time evolution operator, incl. formal solution, as 1 t Z d 1 Û (t, t0 ) = Ĥ(t)Û (t, t0 ) → Û (t, t0 ) = Û (t0 , t0 ) + dt Ĥ(t1 )Û (t1 , t0 ) dt iℏ | {z } iℏ t0 1 =1̂ (4.11) substitute solution into itself 1 t 1 t1 Z  Z  Û (t, t0 ) = 1̂ + dt1 Ĥ(t1 ) 1̂ + dt2 Ĥ(t2 )Û (t2 , t0 ) iℏ t0 iℏ t0 1 t Z Z t Z t1 1 = 1̂ + dt1 Ĥ(t1 ) + dt1 dt2 Ĥ(t1 )Ĥ(t2 )Û (t2 , t0 ) iℏ t0 (iℏ)2 t0 t0...  2 Z t (4.12) 1 t Z Z t1 1 = 1̂ + dt1 Ĥ(t1 ) + dt1 dt2 Ĥ(t1 )Ĥ(t2 ) + · · · iℏ t0 iℏ t0 t0 ∞  m Z t Z tm−1 X 1 = 1̂ + dt1 · · · dtm Ĥ(t1 ) · · · Ĥ(tm ) iℏ t0 t0 m=1  Z t  def. 1 = T exp dt′ Ĥ(t′ ) iℏ t0 so-called time-ordered exponential (via symbol T ) penultimate row dubbed Dyson expansion of Û we have variables t > t1 > · · · > tm > t0 and time-ordered products Ĥ(t1 ) · · · Ĥ(tm ) specifically important if [Ĥ(t), Ĥ(t′ )] ̸= 0, i.e., Ĥ(t)Ĥ(t′ ) ̸= Ĥ(t′ )Ĥ(t) example. if, however, [Ĥ(t), Ĥ(t′ )] = 0 for all t, t′ , then – arbitrary assignment of indices in mth term, ti 7→ tσ(i) , with permutation σ ∗ permutation σ : {1,... , m} → {1,... , m} bijective ∗ set of all permutations of m elements Sm ∗ contains m! distinct reorderings of (1,... , m) to (σ(1),... , σ(m)) – all integrals with permuted indices take same value, meaning Z t Z tm−1 dt1 · · · dtm Ĥ(t1 ) · · · Ĥ(tm ) t0 t0 1 X t Z tσ(m−1) (4.13) Z = dtσ(1) · · · dtσ(m) Ĥ(tσ(1) ) · · · Ĥ(tσ(m) ) m! t0 t0 σ∈S | {z } m =Ĥ(t1 )···Ĥ(tm ) (reordered) 28  4. Perturbation theory – domains of integration ∗ sum over all cases t > tσ(1) > · · · > tσ(m) > t0 , e.g., t1 > t2 (σ leaving index as is) and t2 > t1 (σ swapping indices) for m=2 ∗ zero-measure sets with equality, e.g., t1 = t2 ∗ thus, sum covers—up to a zero-measure volume—the entirety of the m- dimensional cube (t1 ,... , tm ) ∈ [t0 , t]m , e.g., (t1 , t2 ) ∈ [t0 , t] × [t0 , t] – therefore 1 X t Z Z tσ(m−1) dtσ(1) · · · dtσ(m) Ĥ(t1 ) · · · Ĥ(tm ) m! σ∈Sm t0 t0 Z t Z t Z t m (4.14) 1 1 ′ ′ = dt1 Ĥ(t1 ) · · · dtm Ĥ(tm ) = dt Ĥ(t ) m! t0 t0 m! t0 – substitute into Dyson expansion  Z t  ∞  m Z t m 1 ′ ′ X 1 1 ′ ′ T exp dt Ĥ(t ) = 1̂ + dt Ĥ(t ) iℏ t0 iℏ m! t0 m=1 (4.15)  Z t  1 = exp dt′ Ĥ(t′ ) iℏ t0 for the studied case [Ĥ(t), Ĥ(t′ )] = 0 ∀t, t′ 1 t ′ t − t0 Z – furthermore, dt Ĥ(t′ ) = Ĥ for time-independent Hamiltonian iℏ t0 iℏ remark. comments on Dyson expansion – time-ordered result from non-commuting Hamiltonian for different times sim- plifies to known result for time-independent cases – truncation of infinite sum as approximation of Û (t, t0 ) – truncation does not preserve unitarity – Magnus expansion (not further discussed here) as unitary alternative,   Û (t, t0 ) = exp Ω̂1 (t, t0 ) + Ω̂2 (t, t0 ) + · · · , (4.16) with Ω̂j (t, t0 )† = −Ω̂j (t, t0 ) for j ∈ {1, 2,...}, Z t 1 Ω̂1 (t, t0 ) = dt1 Ĥ(t1 ), iℏ t0 Z t Z t1 1 (4.17) Ω̂2 (t, t0 ) = dt1 dt2 [Ĥ(t1 ), Ĥ(t2 )], −ℏ2 t0 t0... 29  4. Perturbation theory 4.3. Interaction picture suppose Ĥ ′ = Ĥ + δ Ĥ as before, together with expected values of an observable L̂ ⟨ψ(t)|L̂|ψ(t)⟩ = ⟨ψ(t)|Û0 Û0† L̂Û0 Û0† |ψ(t)⟩, (4.18) which holds true for all unitarity maps Û0 let |ψint (t)⟩ = Û0 (t, t0 )† |ψ(t)⟩and L̂int = Û0 (t, t0 )† L̂Û0 (t, t0 ), (4.19)   t − t0 with Û0 (t, t0 ) = exp Ĥ , iℏ define the interaction picture of states and observables, respectively using δ Ĥint (t) = Û0 (t, t0 )† δ Ĥ Û0 (t, t0 ), (4.20) we obtain the Schrödinger equation, d    iℏ |ψint ⟩ = Û0† Ĥ + Û0† Ĥ + δ Ĥ |ψ⟩ = δ Ĥint |ψint ⟩ (4.21) dt as well as the Heisenberg equation, ! d ∂ L̂ h i ∂ L̂ iℏ L̂int = −Ĥ Û0† L̂Û + Û0† Û0 + Û0† L̂Û Ĥ = L̂int , Ĥ + , (4.22) dt ∂t ∂t int in the interaction picture remarks. – “intermediate” stage between Heisenberg and Schrödinger picture – time dependency of states via generally time-dependent interaction Hamilton operator δ Ĥint (t) – time dependency of operators via “free” (i.e., unperturbed) contribution Ĥ 4.4. Time-dependent perturbation theory combine Dyson expansion and interaction picture likewise, take equation of motion, d iℏ |ψint (t)⟩ = δ Ĥint (t)|ψint (t)⟩, (4.23) dt and write solution in first-order expansion  Z t  1 ′ ′ |ψint (t)⟩ = T exp dt δ Ĥint (t ) |ψ(t0 )int ⟩ iℏ t0 (4.24) 1 t ′  Z  ′ ≈ 1̂ + dt δ Ĥint (t ) |ψint (t0 )⟩, iℏ t0 propagation of states for time-dependent perturbation in first-order 30  4. Perturbation theory example. transition amplitudes (Fermi’s golden rule) – say initial state |ψint (t0 )⟩ = |En ⟩ eigenstate of unperturbed Hamiltonian Ĥ – goal: find transition amplitude for (projection onto) final state |Em ⟩ – thus (∆t = t − t0 small) 1 t ′  Z  ′ αn→m (∆t) = ⟨Em | 1̂ + dt δ Ĥint (t ) |En ⟩ iℏ t0 1 t ′ Z = δm,n + dt ⟨Em |Û0 (t′ , t0 )† δ Ĥ Û0 (t′ , t0 )|En ⟩ iℏ t0 | {z  } t′ −t0  =exp iℏ [En −Em ] ⟨Em |δ Ĥ|En ⟩  ⟨Em |δ Ĥ|En ⟩ m = n : ∆t = δm,n + iℏ −i[En −Em ]∆t/ℏ − 1
iℏ m ̸= n : En −E m e   ∆t −i∆t[En −Em ]/[2ℏ] En − Em = δm,n + ⟨Em |δ Ĥ|Em ⟩e sinc ∆t , iℏ 2ℏ (4.25) where sinc(x) = sin(x) x and sinc(0) = 1 denotes the cardinal sine function – note that sinc(πx) behaves similarly to δx,0 for integer x – transition probabilities (for m ̸= n) 2 (∆t)2   2 2 En − Em |αn→m (∆t)| = ⟨Em |δ Ĥ|Em ⟩ sinc ∆t (4.26) ℏ2 2ℏ (∆t)2 and |αn→m (∆t)|2 = 1 + ⟨En |δ Ĥ|En ⟩2 for m = n ℏ2 – transition rates   dαn→m (∆t) ⟨Em |δ Ĥ|En ⟩ ∆t = exp [En − Em ] , (4.27) dt iℏ iℏ yielding |dαn→m (∆t)/dt|2 = |⟨Em |δ Ĥ|En ⟩|2 /ℏ2 4.5. Summary perturbative approach to solving problems in quantum physics time-independent Schrödinger equation via perturbed eigenvalues and eigenvectors time-dependent scenarios – Dyson expansion for time-dependent Hamilton operators – interaction picture for separating free and perturbed evolution – first-order approach → transition probabilities/probabilities/rates 31 5. Open quantum systems thus far, closed quantum systems described by a single wave function how to describe an ensemble of states? how do such ensembles evolve when interacting with a bath/the environment? 5.1. Statistical operator consider collection of pure states |ψn ⟩, distinguished by the index n rather than knowing exactly which quantumPstate is realized, we assign probabilities pn , with pn ≥ 0 and n pn = 1 therefore, ensemble of states |ψn ⟩ each of which with a probability pn corresponding statistical operator defined as