APM 421HF/MAT 1723HF: MATHEMATICAL CONCEPTS OF QUANTUM MECHANICS AND QUANTUM INFORMATION THURSDAYS 5 6 (BA 2139), FRIDAYS 5 7 (BA 1240) COURSE NOTES

Transcription

1 APM 42HF/MAT 723HF: MATHEMATICAL CONCEPTS OF QUANTUM MECHANICS AND QUANTUM INFORMATION THURSDAYS 5 6 (BA 239), FRIDAYS 5 7 (BA 240) COURSE NOTES I. M. SIGAL Contents. Week: Structures 2.. State Space 3.2. The Schrödinger Equation 3.3. Conservation of Probability 4.4. Existence of dynamics and self-adjointness 6.5. Supplement: Probability current 6 2. Week 2: Observables The Position and Momentum Operators General Observables Mathematical Supplement: The Fourier Transform Mathematical Supplement: The Self-adjointness of p j 0 3. Week 3: Spectrum and Dynamics 3.. The Spectrum of an Operator 3.2. Spectra of Schrödinger Operators 4 4. Week 4: Physical Significance of Spectrum Spectrum and Measurement Outcomes Space-time Behaviour of Solutions: Bound and Decaying States Mathematical Supplement: Functions of Operators and the Spectral Mapping Theorem Supplement: Sketch of proof of (4.2) 2 5. Week 5: Atoms The Hydrogen Atom Many-particle systems Supplement: Irreducible representations of S n Multi-electron Atoms Week 6: Harmonic Oscillator and Motion in the EM Field The Harmonic Oscillator A particle in an external electro-magnetic field A particle in a constant external magnetic field 32 Date: October, 208.

2 2 I. M. SIGAL 6.4. Linearized Ginzburg-Landau Equations of Superconductivity Supplement: L equivariant functions Supplement: Equivariance and line bundles over complex tori Week 7: Quantum Statistics Density Matrices Quantum statistics: General framework Information Reduction Supplementary material Week 8: Reduced dynamics and quantum channels Reduced dynamics Supplement: Dual quantum evolution Communication channels Measuring classical and quantum information Supplement: Irreversibility Supplement: Stationary States Supplement: Decoherence and Thermalization Week 9: Angular momentum and spin Angular momentum Spin Appendix: Conservation laws Appendix: Motion in a spherically symmetric potential Week 0: Born-Oppenheimer Approximation and Berry Phase Appendix: Projecting-out Procedure 77 References Adiabatic Theory Geometrical Phases Aharonov-Bohm Effect 79. Week: Structures The main ingredients of any physical theory are (a) state space, (b) dynamical law giving evolution of states and (b) the description of the results of measurements. The mathematical description of these structures is extracted from experiments. it is the subject of the next few subsections. The following table provides a summary of these structures and compares them with those in classical mechanics.

3 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 3 Object CM QM state R 3 x R 3 k and L2 (R 3 x) and space Poisson bracket commutator evolution Hamilton s (Newton s) Schrödinger eq is given by eqs observable real function self-adjoint operator on state space on state space results of measurement deterministic probabilistic.. State Space. The space of all possible states of the particle at a given time is called the state space. For us, the state space of a particle will usually be the square-integrable functions: L 2 (R 3 ) := {ψ : R 3 C ψ(x) 2 dx < } R 3 (we can impose the normalization condition as needed). Here R d denotes d- dimensional Euclidean space, R = R, and a vector x R d can be written in coordinates as x = (x,..., x d ) with x j R. The space L 2 (R 3 ) is a vector space, and has an inner-product given by ψ, φ := ψ(x)φ(x)dx. R 3 In fact, it is a Hilbert space (see Section 23. of [7] for precise definitions and mathematical details). What is the physical meaning of ψ(x)? It turns out that ψ(x) 2 is the probability distribution for the particle s position. That is, the probability that a particle is in the region Ω R 3 is Ω ψ(x) 2 dx. Thus we require the normalization ψ(x) 2 dx =. R 3.2. The Schrödinger Equation. Consider a particle in R 3 moving in the potential V (x). As was mentioned already, the evolution of its state is governed by the celebrated Schrödinger equation. If an evolving state at time t is denoted by ψ(x, t), with the notation ψ(t)(x) ψ(x, t), then it satisfies the equation i 2 ψ(x, t) = t 2m xψ(x, t) + V (x)ψ(x, t). (.) Here, = 3 j= 2 j is the Laplace operator, or the Laplacian (in spatial dimension 3), The small constant is Planck s constant; it is one of the fundamental constants in nature. For the record, its value is roughly erg sec.

4 4 I. M. SIGAL The equation (.) can be written as i ψ = Hψ (.2) t where the linear operator H, called a Schrödinger operator, is given by Hψ := 2 ψ + V ψ. (.3) 2m Example.. Here are just a few examples of potentials. () Free motion : V 0. (2) A wall: V 0 on one side, V on the other (meaning ψ 0 here). (3) The double-slit experiment: V on the shield, and V 0 elsewhere. (4) The Coulomb potential : V (x) = α/ x (describes a hydrogen atom). (5) The harmonic oscillator : V (x) = mω2 2 x 2. We will analyze some of these examples, and others, in subsequent chapters. Equation (.2) satisfies certain physically sensible properties: () Causality: The state ψ(t 0 ) at time t = t 0 should determine the state ψ(t) for all later times t > t 0. (2) Superposition principle: If ψ(t) and φ(t) are evolutions of states, then αψ(t) + βφ(t) (α, β constants) should also describe the evolution of a state. (3) Correspondence principle: In everyday situations, quantum mechanics should be close to the classical mechanics we are used to. Proving the first and third properties requires some work. We will do this later on. The second property is equivalent to the fact that Eq (.) (or operator (.3)) is linear. To summarize, with a quantum particle in R 3 moving in the potential V (x), we associate a complex-valued function of position and time, ψ(x, t), x R 3, t R, satisfying the Schrödinger equation (.). ψ(x, t) is called a wave function (or state vector). Moreover, ψ(, t) 2 is the probability distribution for the particle s position. That is, the probability that a particle is in the region Ω R 3 at time t is Ω ψ(x, t) 2 dx. Thus we require the normalization R 3 ψ(x, t) 2 dx =..3. Conservation of Probability. Since ψ(x, t) 2 dx gives the total probability for the particle being somewhere is the space, we should have R 3 ψ(x, t) 2 dx = ψ(x, 0) 2 dx = (.4) R 3 R 3 at all times, t. If (.4) holds, we say that probability is conserved. We prove this property.

5 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 5 First, recall that the evolution of the state (the wave function), ψ, of a particle of mass m in a potential V is given by the Schrödinger equation i ψ = Hψ (.5) t where the linear operator H = 2 + V is the corresponding Schrödinger operator. We supplement equation (.5) with the initial 2m condition ψ t=0 = ψ 0 (.6) where ψ 0 L 2 (R 3 ). The problem of solving (.5)- (.6) is called an initial value problem or a Cauchy problem. It turns out that the property of conservation of probability does not depend on the particular form of the operator H, but rather follow from a basic property symmetry. A linear operator A acting on a Hilbert space H is symmetric if for any two vectors in the domain of A, ψ, φ D(A), Aψ, φ = ψ, Aφ. Theorem.2. Solutions ψ(t) of (.5) with ψ(t) D(H) conserve probability if and only if H is symmetric. Proof. Suppose ψ(t) D(H) solves the Cauchy problem (.5)-(.6). We compute d dt ψ, ψ = ψ, ψ + ψ, ψ = Hψ, ψ + ψ, i i Hψ = [ ψ, Hψ Hψ, ψ ] i (here, and often below, we use the notation ψ to denote ψ/ t). If H is symmetric then this time derivative is zero, and hence probability is conserved. Conversely, if probability is conserved for all such solutions, then Hφ, φ = φ, Hφ for all φ D(H) (since we may choose ψ 0 = φ). This, in turn, implies H is a symmetric operator. The latter fact follows from a version of the polarization identity, ψ, φ = 4 ( φ + ψ 2 φ ψ 2 i φ + iψ 2 + i φ iψ 2 ), (.7) whose proof is left as an exercise below. Homework.3. (Optional) Prove (.7). Homework.4. Show that the following operators on L 2 (R 3 ) (with their natural domains) are symmetric: () x j (that is, multiplication by x j ); (2) p j := i xj ; (3) H 0 := 2 ; 2m (4) the multiplication operator by f(x) (ψ(x) f(x)ψ(x)), provided f : R 3 R is bounded; (5) The Schrödinger operator (.3) (with V (x) real);

6 6 I. M. SIGAL (6) integral operators Kf(x) = K(x, y)f(y) dy with K(x, y) = K(y, x) and, say, K L 2 (R 3 R 3 )..4. Existence of dynamics and self-adjointness. We consider the Cauchy problem (.5)- (.6) for an abstract linear operator H on a Hilbert space H. Here ψ = ψ(t) is a differentiable path in H. Definition.5. We say the dynamics exist if for all ψ 0 H the Cauchy problem (.5)- (.6) has a unique solution which conserves probability. The property of H which guarantees the existence of dynamics is self-adjointness defined as follows Definition.6. A linear operator A acting on a Hilbert space H is self-adjoint if A is symmetric and Ran (A ± i) = H. Note that the condition Ran (A ± i) = H is equivalent to the fact that the equations (A ± i)ψ = f (.8) have solutions for all f H. The definition above differs from the one commonly used but is equivalent to it. This definition isolates the property one really needs and avoids long proofs which are not relevant to us. We have the following key results Theorem.7. The dynamics exist if and only if H is self-adjoint. Theorem.8. Assume that V is real and bounded. Then H := 2 2m + V (x), with D(H) = D( ), is self-adjoint on L 2 (R 3 ). Theorem.9. Let V (x) be a continuous function on R 3 satisfying V (x) 0 and V (x) as x. Then H = 2 2m + V is self-adjoint on L2 (R 3 ) For the proofs of the first two theorems see Sections 2.3 and 2.2 of [7] and a proof of the last one can be found in [8]..5. Supplement: Probability current. We analyze how the probability distribution changes under the Schrödinger equation and derive the differential form of the probability conservation law and formula for the probability current. Proposition.0. The following probability flow equation holds: t ψ 2 = div j(ψ) (.9) where j(ψ) is called the probability current (in the state ψ) and is given by j(ψ) = I(ψ ψ). (.0) m

7 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 7 Proof. We compute t ψ 2 = t (ψψ) = Hψψ + ψ i i div(ψ ψ ψ ψ). i 2m i i Hψ= ψψ + 2m 2m ψ ψ = 2. Week 2: Observables Observables are the quantities that can be experimentally measured in a given physical framework. In this chapter, we discuss the observables of quantum mechanics. 2.. The Position and Momentum Operators. We recall that in quantum mechanics, the state of a particle at time t is described by a wave function ψ(x, t). The probability distribution for the position, x, of the particle, is ψ(, t) 2. Thus the mean value of the position at time t is given by x ψ(x, t) 2 dx (note that this is a vector in R 3 ). If we define the coordinate multiplication operator x j : ψ(x) x j ψ(x) then the mean value of the j th component of the coordinate x in the state ψ is ψ, x j ψ. Using the fact that ψ(x, t) obeys the Schrödinger equation i ψ = Hψ, we t compute d dt ψ, x jψ = ψ, x j ψ + ψ, x j ψ = i Hψ, x jψ + ψ, x j i Hψ = ψ, i Hx i jψ ψ, x j Hψ = ψ, i [H, x j]ψ (where recall [A, B] := AB BA is the commutator of A and B). Since H = 2 2m + V, and (xψ) = x ψ + 2 ψ, we find i [H, x j] = i m j, leading to the equation d dt ψ, x jψ = m ψ, i jψ. As before, we denote the operator i j by p j. As well, we denote the mean value ψ, Aψ of an operator A in the state ψ by A ψ. Then the above becomes m d dt x j ψ = p j ψ (2.) which is reminiscent of the definition of the classical momentum. We call the operator p the momentum operator. In fact, p j is a self-adjoint operator on L 2 (R 3 ). (As usual, the precise statement is that there is a domain on which p j is selfadjoint. Here the domain is just D(p j ) = {ψ L 2 (R d ) x j ψ L 2 (R d ).)

8 8 I. M. SIGAL Note also that, since 2 = p 2 (here p 2 = 3 j= p2 j), we have, for the Schrödinger operator H = 2 2m + V = 2m p 2 + V. Recalling the Hamiltonian function h(x, k) := 2m k 2 + V (x) of Classical Mechanics, we see that the above formula is in agreement with our interpretation of x j and p j as the operators of the j th component of the coordinate and momentum, respectively. Hence, H is often called the quantum Hamiltonian General Observables. Definition 2.. An observable is a self-adjoint operator on the state space L 2 (R 3 ). Homework 2.2. Show that the operators in items ()-(4) in Problem.4 are self-adjoint. (Recall that these operators act on L 2 (R 3 ) and are considered on their their natural domains.) It is shown in Subsection 2.4 that the operators p j (item (2) in Problem.4) is self-adjoint. Later on, we show that, under certain conditions on the potential V (x), the Schrödinger operator (.3) (item (2) in Problem.4) is also self-adjoint. We have already seen two observables: the position operators, x, the momentum operators, p. The Schrödinger operator, H = 2 2m + V, is self-adjoint and therefore, by our definition, is an observable. But what is the physical meaning of this observable? We find the answer below. In general, we interpret A ψ as the average of the observable A in the state ψ. The reader is invited to derive the following equation for the evolution of the mean value of an observable. Homework 2.3. Check that for any observable, A, and for any solution ψ of the Schrödinger equation, we have d dt A ψ = i [H, A] ψ. (2.2) This result is analogous to the classical evolution hamiltonian equations, d dt a = {h, a}, for classical observables, a, i.e. functions on the phase space. Consequently, we interpret H as the observable of energy or quantum Hamiltonian, or the energy or Hamiltonian operator.

9 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 9 Now, we would like to apply (2.2) on the momentum operator. Simple computations give [, p] = 0 and [V, p] = i V, so that i [H, p] = V and hence d dt p j ψ = j V ψ. (2.3) This is a quantum mechanical mean-value version of Newton s equation of classical mechanics. Or, if we include Equation (2.), we have a quantum analogue of the classical Hamilton or Newton s equations (Ehrenfest theorem). Finally, by analogy with above, we introduce the angular momentum operators, L j = (x p) j, with L := (L, L 2, L 3 ), where p = i as above. The operatorvector L can be written as L = x p. This is the angular momentum operator, or angular momentum observable, or just angular momentum Using the Fourier transform (see the supplementary subsection below), we compute the mean value of the momentum operator ψ, p j ψ = ˆψ, p j ψ = ˆψ, k j ˆψ = R 3 k j ˆψ(k) 2 dk. This, and similar computations, show that ˆψ(k) 2 is the probability distribution for the particle momentum Mathematical Supplement: The Fourier Transform. The Fourier transform is a powerful tool in many areas of mathematics and physics. Here we list its main properties. For more details see Section 23.4 of the Mathematical Supplement in [7]. The Fourier transform is a map, F, which sends a function ψ : R d C to another function ˆψ : R d C where for x, k R d, ˆψ(k) := (2π ) d/2 R d e ik x/ ψ(x)dx (it is convenient for quantum mechanics to introduce Planck s constant,, into the Fourier transform). We first observe that ˆψ = Fψ is well-defined if ψ is an integrable function (ψ L (R d ), meaning R d ψ(x) dx < ), and that F acts as a linear operator on such functions. Homework 2.4. Show that F is a bounded, linear operator from L (R d ) to L (R d ). The following exercise asks to calculate a very important Fourier transform of the Gaussian. Homework 2.5. Show that under F () e x 2 2a 2 ( a) d/2 e a k 2 2 (Re(a) > 0). Hint: try d = first complete the square in the exponent and move the contour of integration in the complex plane.

10 0 I. M. SIGAL (2) e 2 2 x A x d/2 (det A) /2 e 2 k Ak (A a positive d d matrix). Hint: diagonalize and use the previous result. In the first problem, since Re(a) > 0, the function on the left is in L (R d ), and the Fourier transform is well-defined. However, we can extend this result to Re(a) = 0, in which case the integral is convergent, but not absolutely convergent. The utility of the Fourier transform derives from the following properties: () i x ψ(k) = k ˆψ(k); (2) xψ(k) = i k ˆψ(k). (3) ˆψ, ˆφ = ψ, φ. The first two properties can be loosely summarized by saying that the Fourier transform exchanges differentiation and coordinate multiplication. The third property implies that (a) F is an isometry, Fu = u, (b) F can be extended to a bounded map on L 2 (R d ) and (c) F has an inverse, F, i.e. there is a bounded operator, denoted F, s.t. F F = and FF =. A bounded operator, U, obeying Uψ, Uφ = ψ, φ is said to be unitary. Homework 2.6. Prove properties () and (2) Mathematical Supplement: The Self-adjointness of p j. Proposition 2.7. The operators p j = i xj acting on the space L 2 (R 3 ) (with the domain H (R 3 )) are self-adjoint. Proof. To simplify notation, we consider the one dimensional case: the operator p = i x acting on the space L 2 (R) (with the domain H (R)) and show that it is self-adjoint. Indeed, the operator p is symmetric, so it suffices to show that Ran ( i x +i) = L 2 (R). The latter property is equivalent to the statement that the equation ( i x + i)ψ = f, has a unique solution for every f L 2 (R). We solve this equation using the Fourier transform (see Section 2.3). Applying the FT to this equation, we find that it is equivalent to the equation (k + i) ˆψ(k) = ˆf(k), which can be explicitly solved ˆψ(k) = ˆf(k) k + i Now for any f L 2 (R), we have Property (3) of the FT that ˆf L 2 (R) and therefore k + i ˆψ(k) = ˆf(k) L 2 (R),

11 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION Since k + i = ( + k 2 ) /2, we see that ˆψ L 2 (R) and k ˆψ L 2 (R). Again by Property (3) of the FT, this implies that ψ L 2 (R) and ψ L 2 (R), so ψ lies in the Sobolev space of order one, H (R) := {ψ L 2 (R) : ψ L 2 (R)}, which is the domain, D( i x ), of the operator p. Hence Ran ( i x + i) = L 2. Similarly Ran ( i x i) = L Week 3: Spectrum and Dynamics Given a quantum observable (a self-adjoint operator) A, what are the possible values A can take in various states of the system? How to classify the dynamics without solving the Schrödinger explicitly? These two questions will be addressed and answered in this section and the key to the answers is the notion of the spectrum. We begin by presenting the general theory, and then proceed to applications. Details about the general machinery can be found in Section 23.8 of [7]. 3.. The Spectrum of an Operator. We begin by giving some key definitions and statements related to the spectrum. We begin with some definitions. We say an operator A on a Hilbert space H is invertible invertible iff there is a bounded operator B s.t. BA =, AB =. The operator B is called the inverse of A and is denoted A. Homework 3.. Show that if operators A and C are invertible, and C is bounded, then the operator CA is defined on D(CA) = D(A), and is invertible, with (CA) = A C. In what follows, we consider operators of the form A λ and we will often omit the identity operator writing instead A λ. Definition 3.2. The spectrum of an operator A on a Hilbert space H is the subset of C given by σ(a) := {λ C A λ is not invertible (has no bounded inverse)}. The following exercise asks for the spectrum of our favorite operators. Homework 3.3. Prove that as operators on L 2 (R d ) (with their natural domains), () σ() = {}. (2) σ(p j ) = R. (3) σ(x j ) = R. (4) σ(v ) = Ran (V ), where V is the multiplication operator on L 2 (R d ) by a continuous function V (x) : R d C. (5) σ( ) = [0, ).

12 2 I. M. SIGAL (6) (optional) σ(f(p)) = Ran (f), where f(p) := F ff with f(k) the multiplication operator on L 2 (R d ) by a continuous function f(k) : R d C. The following three results state important general facts about the spectrum. Theorem 3.4. The spectrum σ(a) C is a closed set. Theorem 3.5. A is bounded iff the spectrum σ(a) is a bounded set. Theorem 3.6. A self-adjoint = σ(a) R. Proofs of these results rely on the following simple but very powerful theorem: Theorem 3.7 (Perturbation expansion). Assume the operator T is invertible, and the operator V is bounded and satisfies V T <. Then the operator T + V, defined on D(T + V ) = D(T ), is invertible. Moreover, its inverse is given by the following absolutely convergent series (T + V ) = T ( V T ) n. n=0 Homework 3.8. Prove this theorem. (Hint: Show that, under the assumptions of the theorem, the series n=0 T ( V T ) n, called a Neumann series, is absolutely convergent (i.e. n=0 T ( V T ) n < ) and provides the inverse of the operator T + V.) The complement of the spectrum of A in C is called the resolvent set of A: ρ(a) := C\σ(A). For λ ρ(a), the operator R A (λ) := (A λ), called the resolvent of A, is well-defined and bounded. Proof of Theorem 3.4. Showing that σ(a) is closed is equivalent to showing that the resolvent set, ρ(a), is an open set. Let z ρ(a). Then A z is invertible. Write A z = A z + z z. Then, by Theorem 3.7 (with T = A z and V = (z z )), A z is invertible, if z z < (A z). Proof of Theorem 3.5. We prove only the direction: if A is bounded iff then the set σ(a) is bounded. By Theorem 3.7 (with T = z and V = A), the operator z A is invertible, if Az <, i.e. z > A. Hence any z with z > A is in ρ(a) and therefore σ(a) {z C : z A }. We will use the following criterion of invertibility: an operator A is invertible iff it is one-to-one and onto, that is, Null A = {0} and Ran A = H. (We omit the proof.) Here Null A is the kernel or nullspace of A: Null A := {u D(A) Au = 0}. (3.) Proof of Theorem 3.6. Write z = λ + iµ with λ, µ R. Then, since A is symmetric, we have (A z)u 2 = (A z)u, (A z)u = (A λ)u 2 + µu 2 µ 2 u 2. (3.2)

13 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 3 Hence, if Iz 0, then Null(A z) = { 0 }. Now, by the second property of self-adjoint operators, Ran (A ± i) = H. Hence by above A ± i are invertible. Now, let A z be invertible for some z C/R. Then, taking v := (A z)u and u(a z) v in (3.2), we have the estimate (A z) Iz. (3.3) By Theorem 3.7, A z = (A z)+(z z ) is invertible for z z < (A z). Therefore, by (3.3), A z is invertible if z z < Iz. This way starting with either z = i or z = i, we can extend invertibility of A z to all Iz > 0 (or Iz < 0). This shows ρ(a) C/R and therefore σ(a) R. In addition we have proven Lemma 3.9. Let A be a self-adjoint operator. Then, for every z C/R, (A z) is invertible and the inverse satisfies bound (3.3) Recall that an operator A is invertible iff it one-to-one and onto: that is, Null A = {0} and Ran A = H. Thus the failure one of one these conditions for A z assures that z σ(a). In particular, if Null(A z) is nontrivial, then z is an eigenvalue of A. The integer dim Null(A z) is called the multiplicity of z. This leads to the definition. The discrete spectrum of an operator A is σ d (A) = {λ C λ is an isolated eigenvalue of A with finite multiplicity} (isolated meaning some neighbourhood of λ is disjoint from the rest of σ(a)). Homework 3.0. () Show Null(A λ) is a vector space. (2) Show that if A is self-adjoint, eigenvectors of A corresponding to different eigenvalues are orthogonal. The rest of the spectrum is called the essential spectrum of the operator A: σ ess (A) := σ(a)\σ d (A). Remark 3.. Some authors may use the terms point spectrum and continuous spectrum rather than (respectively) discrete spectrum and essential spectrum. Homework 3.2. For the following operators on L 2 (R d ) (with their natural domains), show that () σ ess (p j ) = σ(p j ) = R; (2) σ ess (x j ) = σ(x j ) = R; (3) σ ess ( ) = σ( ) = [0, ). Hint: Show that these operators do not have discrete spectrum.

14 4 I. M. SIGAL Homework 3.3. Let A be a self-adjoint operator on H. If λ is an accumulation point of σ(a), then λ σ ess (A). Hint: use the definition of the essential spectrum, and the fact that the spectrum is a closed set Spectra of Schrödinger Operators. We now want to address the question of how to characterize the essential spectrum of a self-adjoint operator A. Is there a characterization of σ ess (A) similar to that of σ d (A) in terms of some kind of eigenvalue problem? In particular, we address this question for Schrödinger operators. We begin with Definition 3.4. Let A be an operator on L 2 (R d ). A sequence {ψ n } D(A) is called a spreading sequence for A and λ if () ψ n = for all n (2) for any bounded set B R d, supp(ψ n ) B = for n sufficiently large (3) (A λ)ψ n 0 as n. The second condition rules out λ σ d (A). If not for it, a sequence consisting of a repeated eigenfunction would fit this definition. The second condition implies that we can choose a subsequence {ψ n} so that supp ψ n { x R n }, with R n. In what follows we always assume that the sequence {ψ n } has this property. As an example, a spreading sequence for 2 and any λ > 0 is given by 2m ψ n (x) := f( x /n)e ik x/, where f is any smooth function, s.t. f(x) = for x and = 0 for x 2, and 2m k 2 = λ. Homework 3.5. Show that the sequence ψ n (x) := n 3/2 f(x/n)e ik x/, where f is any smooth function, s.t. f(x) = 0 if either x /2 or x and f(x) 2 dx =, is a spreading sequence for 2 and λ = 2m 2m k 2. We proceed to Schrödinger operators, H = 2 + V. By Theorems.8 and 2m.9, they are self-adjoint if V (x) is real and either bounded or continuous and satisfies V (x) 0 and V (x) as x. Theorem 3.6 (Weyl-Zhislin theorem). Let H = 2 + V be a Schrödinger 2m operator with real potential V (x) which is continuous and bounded from below. Then σ ess (H) = {λ there is a spreading sequence for H and λ}. A detailed sketch of the proof of this result can be found in [7], Section 6.3. Now we describe the spectrum σ(h) of the self-adjoint Schrödinger operator H = 2 + V. Our first result covers the case when the potential tends to zero 2m at infinity.

15 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 5 Theorem 3.7. Let V : R d R be continuous, with V (x) 0 as x. Then σ ess (H) = [0, ) (so H can have only negative isolated eigenvalues, possibly accumulating at 0). Proof. Denote H 0 := 2. We have, by the triangle inequality, 2m (H λ)ψ n V ψ n (H 0 λ)ψ n (H λ)ψ n + V ψ n. Suppose {ψ n } is a spreading sequence. Then the term V ψ n goes to zero as n because V goes to zero at infinity and {ψ n } is spreading. So λ is in the essential spectrum of H if and only if λ σ ess (H 0 ). We have (see Problem 3.2) σ ess (H 0 ) = σ ess ( ) = [0, ), and consequently σ ess (H) = [0, ). The bottom, 0, of the essential spectrum, is called the ionization threshold, since above this energy the particle is not longer localized, but moves freely. Our next theorem covers confining potentials that is, potentials which increase to infinity with x. Schrödinger operators with such potentials are self-adjoint (the proof of this fact is fairly technical, it can be found, for example, in [8].) Theorem 3.8. Let V (x) be a continuous function on R d satisfying V (x) 0, and V (x) as x. Then σ(h) consists of isolated eigenvalues {λ n } n= with λ n as n. Proof. Denote H 0 := 2 2m, so that H = H 0 + V. Suppose λ is in the essential spectrum of H, and let {ψ n } be a corresponding spreading sequence. Then as n, 0 ψ n, (H λ)ψ n = ψ n, H 0 ψ n + ψ n, V ψ n λ = 2 ψ n 2 + V ψ n 2 λ 2m inf V (y) λ y supp (ψ n) (because {ψ n } is spreading), which is a contradiction. Thus the essential spectrum is empty. Now, assume that H had a finite number of eigenvalues. Then its spectrum would be bounded and therefore by Theorem 3.5, the operator H itself would be bounded, which is a contradiction, as we know that H is unbounded. Hence H has an infinite number of eigenvalues. Since the essential spectrum of H is empty, these eigenvalues must have finite multiplicities and accumulate at +. Homework 3.9. What are the essential spectra and what are possible locations of the discrete spectra of the following operators (justify your answer) (a) H = 2 2m 0 x 3 + x 4, (b) H = 2 ( + 2m x ) 2.

16 6 I. M. SIGAL 4. Week 4: Physical Significance of Spectrum Here we address two issues: () Spectrum and measured values of quantum observables; (2) Spectrum and space-time behaviour of wave functions. 4.. Spectrum and Measurement Outcomes. In Quantum Mechanics the probability that in the state ψ, the physical quantity corresponding to the observable A is in a set Ω as Prob ψ (A Ω) = ψ, χ Ω (A)ψ (4.) Here χ Ω (λ) is the characteristic function of the set Ω (i.e. χ Ω (λ) =, if λ Ω and χ Ω (λ) = 0, if λ / Ω) and the operator χ Ω (A) can be defined using an operator calculus. We define it in important special cases below. For A self-adjoint, the operator χ Ω (A) is also self-adjoint. We call it the probability observable. This definition is validated by quantum experiments. Mathematically, it is a projection, i.e. it satisfies χ Ω (A) 2 = χ Ω (A). Our goals are () to define the operator χ Ω (A); (2) to show that χ Ω (A) = χ Ω σ(a) (A), (4.2) χ Ω (A) 0 Ω σ(a). (4.3) The second property shows that the spectrum is equal to the set of all the values the physical quantity represented by the observable A could take, or in short σ(a) = {values of the physical quantity represented by A}. (4.4) Due to the relation (4.), the above equation suggests that σ(a) can be interpreted as the set of all possible values of the observable A. The most important observable is the energy the Schrödinger operator, H, of a system. Hence the spectrum of H gives the possible values of the energy. Now, assume Ω C is a bounded and open domain. We observe that by Cauchy s residue theorem of the complex analysis, we have χ Ω (λ) := 2πi γ (z λ) dz (4.5) where γ = Ω. Now, assume that Ω ρ(a). Then the following operator is well defined χ Ω (A) := (A z) dz (4.6) 2πi γ The integral here can be understood either as the Bochner integral of vector-valued functions with values in a Banach space, i.e. as a Banach space limit Riemann

17 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 7 sum approximations, or in the week sense as follow. For any ψ, φ H, we define the l.h.s. by the r.h.s.as ψ, χ Ω (A)φ := ψ, (A z) φ dz 2πi γ (knowledge of ψ, T φ for all ψ and φ determines the operator T uniquely). We call χ Ω (A) the characteristic function of the operator A. This definition coincides with all other definitions for χ Ω (A). We claim that with this definition we have (4.2) (of course, assuming for simplicity the restriction Ω ρ(a)). Indeed, if Ω σ(a) =, then the resolvent (A z) is analytic in Ω and therefore by another Cauchy s theorem γ (A z) dz = 0 and therefore χ Ω (A) = 0 which implies the desired statement Space-time Behaviour of Solutions: Bound and Decaying States. Consider the self-adjoint Schrödinger operator, H = 2 +V, acting on 2m L2 (R 3 ). We show how the classification of the spectrum introduced in the previous section is related to the space-time behaviour of solutions of the Schrödinger equation i ψ = Hψ (4.7) t with given initial condition ψ t=0 = ψ 0. (4.8) Naturally, we want to distinguish between states which are localized for all time, and those whose essential support moves off to infinity. We assume all functions below are normalized. We assume that the potential V is real and bounded and such that H has no eigenvalues embedded in its essential spectrum. We define the subspaces of the discrete and essential spectra H discr := { span of eigenfunctions of H}, (4.9) H ess := { span of eigenfunctions of H}, (4.0) where W := {ψ H ψ, w = 0 w W }. By the definition, H discr H ess = H L 2 (R 3 ). We have the following key result. Theorem 4.. (i) If ψ 0 H discr, then for any ɛ > 0, there is an R such that inf ψ 2 ɛ. (4.) t x R (ii) If ψ 0 H ess, then for any R, ψ 2 0 (4.2) x R as t, in the sense of ergodic mean.

18 8 I. M. SIGAL Convergence f(t) 0 in ergodic mean as t means that T T 0 f(t)dt 0 as T. Under some conditions on V one can upgrade (4.2) to the uniform convergence. The proof of (4.) is elementary and is given below. The second result is called the RAGE theorem. Its proof is rather involved and is sketched in Supplement 4.4 (see, eg, [3, 8] for a complete proof). A solution, ψ, of the SE satisfying (4.) is called a bound state, as it remains essentially localized in space for all time. A solution, ψ, satisfying (4.2) is called a decaying state, as it eventually leaves any fixed ball in space. We say that a system in a bound state is stable, while in a decaying state, unstable. This notion differs from the notion of stability in dynamical systems. The notion of stability here characterizes the space-time behaviour of the system, whether it stays essentially in a bounded region of the space, or falls apart with fragments departing from each other. Now, we saw above that solutions of the Schödinger equation with initial conditions in the discrete spectral subspace describe bound states, those in the essential spectral subspace describe decaying states. Hence the spectral classification for a Schrödinger operator H leads to a space-time characterization of the quantum mechanical evolution, ψ. Namely, the classification of the spectrum into discrete and essential parts corresponds to a classification of the dynamics into localized (bound) states and locally-decaying (scattering) states. Proof of Theorem 4.(i). We begin with some elementary but important statements. Proposition 4.2. if Hψ j = λ j ψ j, then the solution of the SE (4.7) with the initial condition ψ j is given by ψ = e iλ j t ψ j. Proof. Indeed, the latter function satisfies (4.7) with the initial condition ψ j. On the hand, by the conservation of probability the solution of the Schrödinger initial value problem is unique, as shown in Proposition 4.3 below. Proposition 4.3. If H is symmetric, then the solution of the Cauchy problem (4.7)-(4.8) is unique. Proof. Assume on contrary there are two solutions, ψ and φ, of the Cauchy problem (4.7)-(4.8). Then χ := ψ φ solves (4.7) with the initial condition 0. By Theorem.2, the conservation of probability formula (.4) says χ(x, t) 2 dx = R 3 χ(x, 0) 2 dx = 0 and therefore χ = 0. So the solution of the Schrödinger initial R 3 value problem is unique. Now let ψ 0 H discr normalized as ψ 0 =, and {ψ j } be an orthonormal set ( ψ j, ψ k = δ jk ) of eigenfunctions of H: Hψ j = λ j ψ j. Then ψ 0 can be written

19 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 9 as ψ 0 = j a jψ j where a j C, j a j 2 =. Define ψ0 N = N j= a jψ j. For any ɛ > 0 we choose N so that ψ 0 ψ0 N ε/2. Let ψ and ψ N solve (4.7) with the initial conditions ψ 0 and ψ0 N. By the conservation of probability formula (.4), we have ψ ψ N = ψ 0 ψ N 0 ε/2. (4.3) Furthermore, by Proposition 4.2 and the linearity of (4.7), ψ N can be written as N ψ N = e iλjt/ a j ψ j. j= Multiplying this equation by the characteristic function of the exterior of the R ball, x R, taking the L 2 norm and using the triangle inequality, we obtain ( x R ψ N 2 ) /2 N j= ( ) /2 a j ψ j 2. (4.4) x R To estimate the second factor on the right hand side, for any ε > 0, we choose R such that for all j, ( ) /2 ψ j 2 ε 2 N. x R Using this estimate and applying the Cauchy-Schwarz inequality to the sum on the right hand side of (4.4) and using that N j= = N and j a j 2 =, we obtain ( x R ψ N 2 ) /2 ε/2. This inequality together with (4.3) gives (4.) Mathematical Supplement: Functions of Operators and the Spectral Mapping Theorem. Our goal in this section is to define functions f(a) of a self-adjoint operator A. We do this in the special case where A is bounded and therefore so is σ(a), and f is a function analytic in a neighbourhood of σ(a). Suppose f(λ) is analytic in a complex disk of radius R, {λ C λ < R}, with R > A. So f has a power series expansion, f(λ) = n=0 a nλ n, which converges for λ < R. Define the operator f(a) by the convergent series f(a) := a n A n. (4.5) n=0 We have already encountered an example of this definition: the function f(λ) = (λ z),

20 20 I. M. SIGAL for z > A, which is analytic in a disk of radius R, with A < R < z, and has power-series expansion f(λ) = z λ/z = (λ/z) j, z j=0 for λ < z. The corresponding operator defined by (4.5) is the resolvent (A z) = z j A j. (4.6) z j=0 (Recall that the series in (4.6) is a Neumann series.) Recall that the resolvent, (A z), is defined for any z in the resolvent set ρ(a) = C\σ(A) and is an analytic (operator-valued) function of z ρ(a). Suppose f is analytic in a complex neighbourhood of σ(a). We can replace definition (4.5) by the contour integral (called the Riesz integral) f(a) := f(z)(a z) dz (4.7) 2πi Γ where Γ is a contour in C encircling σ(a). The integral here can be understood in the following sense: for any ψ, φ H, ψ, f(a)φ := f(z) ψ, (A z) φ dz 2πi Γ (knowledge of ψ, f(a)φ for all ψ, φ determines the operator f(a) uniquely). Homework 4.4 (Optional). Show that the definition (4.7) agrees with (4.5) when f(λ) is analytic on { λ < R} with R > A. Hint: by the Cauchy theorem, and analyticity of the resolvent, the contour Γ can be replaced by { z = R 0 }, A < R 0 < R. On this contour, (A z) can be expressed as the Neumann series (4.6). A similar formula can be used for unbounded operators A to define certain functions f(a) (see [8]). The following useful result relates eigenvalues and eigenfunctions of A to those of f(a). Theorem 4.5 (Spectral mapping theorem). Let A be a bounded operator, and f a function analytic on a disk of radius > A. If Aφ = λφ, then f(a)φ = f(λ)φ. Moreover, σ(f(a)) = f(σ(a)). Proof. If {a j } are the coefficients of the power series for f, then f(a)φ = a j A j φ = ( a j λ j )φ = f(λ)φ. j=0 j=0 To prove the second statement one shows that (f(a) f(z)) is invertible iff so is (A z).

21 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION Supplement: Sketch of proof of (4.2). in this proof we display the time dependence as a subindex t. We suppose the potential V (x) is bounded, and so as a multiplication operator V = V, and therefore V V. Since also 0, the operator H is bounded below: H V. Let λ > V + so that H + λ >, and let φ t := (H + λ)ψ t = e iht (H + λ)ψ 0. Defining B := ( + λ)(h + λ), (4.8) and using that B = V (H + λ), which shows that B is bounded, we see that ψ t = ( + λ) Bφ t. Let χ Ω denote the characteristic function of a set Ω R 3 and let K(x, y) be the integral kernel of the operator χ Ω ( + λ). Using (4.8) and the notation K x (y) := K(x, y), we obtain that χ Ω ψ t = χ Ω ( + λ) Bφ t = K x, Bφ t = B K x, φ t. (4.9) We use the notation x for the L 2 norm in the variable x and claim now that T T 0 dt B K x, φ t 2 x 0, as T. (4.20) To prove this claim we use the fact that K(x, y) 2 dxdy < to show that B K x, φ t B K x φ t B K x (H + λ)ψ 0 L 2 (dx) (uniformly in t). Next, we want to prove that x, T T 0 dt B K x, φ t 2 0, as T. (4.2) Then (4.20) follows from interchange of t and x integration on the l.h.s. and the Lebesgue dominated convergence theorem. The proof of (4.2) is a delicate one. First note that, since ψ 0 the eigenfunctions of H, so is φ 0 := (H + λ)ψ 0 (show this). Next, we write f, φ t 2 = f f, φ t φ t = f f, e itl/ φ 0 φ 0. (4.22) where L is an operator acting on H H given by L := H H. The relation φ 0 (the eigenfunctions of H) implies that φ 0 φ 0 (the eigenfunctions of H H). Hence we can compute the time integral of the r.h.s.: T T 0 dt f f, e itl/ φ 0 φ 0 = T f f, e it L/ φ 0 il/ φ 0. (4.23) The delicate point here is to show that, for nice f and φ 0, the r.h.s. is welldefined and is bounded by CT δ, δ > 0. We omit showing this here (this can be done, for example. by using the spectral decomposition theorem, see []). Then (4.2) holds, which completes the proof (since this shows that for any bounded set Ω R 3 we have that χ Ω ψ t 0, in the sense of ergodic mean, as t, which is equivalent to (4.2)).

22 22 I. M. SIGAL 5. Week 5: Atoms 5.. The Hydrogen Atom. A hydrogen atom consists of a proton and an electron, interacting via a Coulomb force law. Separating the centre of mass or assuming for simplicity that the nucleus (the proton) is infinitely heavy (m p /m e 836) and placing it at the origin, we have the electron moving under the influence of the external (Coulomb) potential V (x) = e 2 / x, where e is the charge of the proton, and e that of the electron. The appropriate Schrödinger operator is therefore H = 2 2m q2 x acting on the Hilbert space L 2 (R 3 ). Here q = e/ 4πɛ 0 (ɛ 0 is the permittivity of vacuum). By an extension of Theorem 3.7 to unbounded potentials, we conclude that H is self-adjoint and σ ess (H) = [0, ). Hence the discrete spectrum, corresponding to bound states, could lie only on negative semi-axis. There could be finite or infinite number of eigenvalues, or none. Thus our goal is to find the discrete spectrum The hydrogen atom is one of three cases when this can be done explicitly (the other two are the infinite well and the harmonic oscillator.) We will not go into the computations but just quote the result: the eigenvalues are ( ) mq 4 E n = 2 2 n, (5.) 2 with the degeneracies n 2. (The universal constant (fine-structure constant) α := e 2 / c /37.) The ground state energy is E = mq 4 /2 2 and is non-degenerate, as also follows from a general argument. So we see that the hydrogen atom has an infinite number of bound states below the essential spectrum (which starts at zero), which accumulate at zero. This result can be obtained by a straightforward extension of a general technique developed in Subsection?? below without solving the eigenvalue problem. Finally, since the energy of the emitted photon is ω = E i E f, where E f < E i are the final and initial energies of the atom in a radiation process, we see that the expression (5.) is in a complete agreement with the empirical formula for the wavelength, λ, of emitted radiation ( Balmer series ) λ = R( n 2 f ). n 2 i Here n f < n i are integers labeling the final and initial states of the atom in a radiation process, R is a constant. The latter formula predates quantum

23 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 23 mechanics, and was derived from measurements of absorption and emission spectra and meaning of integers, n f < n i, was unknown at the time Many-particle systems. Now we consider a physical system consisting of n particles of masses m,..., m n which interact pairwise via the potentials v ij (x i x j ), where x j is the position of the j-th particle. Examples of such systems include atoms or molecules i.e., systems consisting of electrons and nuclei interacting via Coulomb forces. In classical mechanics such a system is described by the particle coordinates, x j and momenta k j, j =,..., n, so that the classical state of the system is given by the pair (x, k) where x = (x,..., x n ) and k = (k,..., k n ) and the state space, also called the phase-space, of the system is R 3n x R 3n k, or a subset thereof. The dynamics of this system is given by the classical Hamiltonian function h(x, k) = n j= 2m j k j 2 + V (x) where V is the total potential of the system, and the standard Poisson brackets Since in our case the particles interact only with each other and by two-body potentials v ij (x i x j ), the total V is given by V (x) = v ij (x i x j ). (5.2) 2 i j As in the one-particle case, we define the state space as L 2 (R 3n ). Elements of this space are complex functions Ψ(x,..., x n ) L 2 (R 3n, C). (5.3) Furthermore, we associate with particle coordinates x j, and momenta k j, the quantum coordinates x j, and momenta p j := i xj, which are operators. And so the classical Hamiltonian h(x, k) leads to the Schrödinger operator H n := h(x, p), p = (p,..., p n ), i.e. H n = n j= 2m j p j 2 + V (x), (5.4) acting on L 2 (R 3n ). This is the Schrödinger operator, or quantum Hamiltonian, of the n particle system. Identical particles. Quantum many-particle systems display a remarkable new feature. Unlike in classical physics, identical particles (i.e., particles with the same masses, charges and spins, or, more generally, which interact in the same way) in quantum physics are indistinguishable in quantum physics. Assume we have n identical particles (of spin 0). Classically, states of such a system are given by (x, k,..., x n, k n ), with the state space, also called the phase-space, being

24 24 I. M. SIGAL n (R 3 x R 3 k ) = R3n x quantum system is R 3n k. Naively, we might assume that the state space of the L 2 (R 3n, C) n L 2 (R 3, C), (5.5) The indistinguishability of the particles means that all probability distributions which can be extracted from an n-particle wave function Ψ(x,..., x n ), should be invariant with respect to permutations of the coordinates, x j, of identical particles. To illustrate this, we consider bound states which can be always taken to be real wave functions. Since Ψ(x,..., x n ) 2 is invariant under permutations, this implies that Ψ(x,..., x n ) is invariant under particle permutations, modulo a change of sign. Since all particle permutations of n indices form a group (the symmetric group, S n ), this imposes constraints on how Ψ(x,..., x n ) transforms. To explain this, we need some definitions. Consider one-to-one and onto maps, π, of {, 2,..., n} into itself, called a permutation. With a permutation, we associate the transformation of R 3n,also denoted by π, defined as π(x,..., x n ) = (x π(),..., x π(n) ). We consider the map of S n by unitary (i.e. preserving the inner product) operators on L 2 (R 3n ) defined by (T π Ψ)(x) = Ψ(π x), with the property T π T π2 = T π π 2 (this is why we need π on the r.h.s.), called the (unitary) representation of S n. Let U(X) denote the space of unitary operators acting on a Hilbert space X. For a group G, a representation T : G U(X) is called irreducible iff X has no non-trivial proper subspace invariant under all T g, g G. The dimension of T is the dimension of X. There are exactly two one-dimensional irreducible representations of S n and they are of the special interest in the spinless case: where Ψ(x,..., x n ) are totally invariant under permutations, T π Ψ = Ψ π S n, and where Ψ(x,..., x n ) transform as T π Ψ = ( ) #(π) Ψ π S n, (5.6) where #(π) is the number of transpositions making up the permutation π (so ( ) #(π) is the parity of π S N ). (See Supplement 5.3 for more details on irreducible representations of S n.) The particles described by the wave functions of the first type are called bosons and by the second type, fermions. Correspondingly, we have the following subspaces of L 2 (R 3n ): H total bose := {Ψ L 2 (R 3n, C) T π Ψ = Ψ}, (5.7) H total fermi := {Ψ L 2 (R 3n, C) T π Ψ = ( ) #(π) Ψ} (5.8)

25 APM 42HF/MAT 723HF: QUANTUM MECHANICS AND QUANTUM INFORMATION 25 The Schrödinger operators describing identical bosons or fermions act on these spaces. We consider an important example, illustrating the dramatic effect the choice of the subspace has on the spectrum. Ideal quantum gas. Ideal quantum gas (a system of n identical particles not interacting mutually) in an external potential V (x). The same as above but with v = 0. Hence, the Schrödinger operator for such a system is H igas = n j= ( 2 2m x j + U(x j )). (5.9) Assume the Schrödinger operator 2 2m x +U(x) has an isolated eigenvalue at the bottom of its spectrum - the ground state energy. In this case, so does H idealgas and, for bosons, its ground state and ground state energy are given by Φ(x,..., x n ) = φ(x )... φ(x n ), and E (n) = ne, (5.0) respectively, where where φ(x) and e are the ground state and the ground state energy of the Schrödinger operator 2 2m x + U(x). Homework 5.. Prove the above statement. On the other hand, for spinless fermions after separation of spin variables and for n 2, the ground state of the operator H igas cannot be given by the product n ϕ (x j ), where ϕ (x) is the ground state of the operator 2 2m x + U(x). A little contemplation shows that it is given by the anti-symmetric product, n φ j (x j ), of n bound states, φ j (x), of the operator H igas, corresponding to the n lowest energies, say e,..., e n. (The case of n odd requires a slight modification.) In this case, the ground state energy, E (n), of H igas is given by E (n) = n e j Homework 5.2. Prove the above statement. This is, in general, a much larger number. Take for instance the Hydrogen atom (V (x) = e2 ). Then the energies E x j behave roughly as m(e2 Z) 2 j 2 (see 2 2 (5.)) and E (n) = O(), rater than E (n) = O(n) in the bosonic case. Say, the number of atoms in the room is of order n = 0 25, so we see that the difference is considerable. It is not easy to give a realistic estimate of the sum n E j. There is considerable activity in producing such estimates (in terms of V ), which are called the Lieb-Thirring inequalities.

26 26 I. M. SIGAL 5.3. Supplement: Irreducible representations of S n. Consider partitions, α, of the integer n into ordered positive integers 2r + α α 2 α k. Denote the set of such α s by A r. These can be visualized as arrangements of n squares into k columns having α, α 2,..., α k squares each, which are stuck side by side, and which are called Young diagrams (we retain for the them the same notation α), see the figure below. In particular, for spin one half, r =, we have one- and two-column Young 2 diagrams. With a Young diagram α one associates a Young tableau, by filling in squares with particles. We associate with a given Young diagram, α, the space, H α, of wave functions which are symmetric with respect to permutations of particles in the same row and antisymmetric with respect to those in the same column of some T (say, canonical one), associated with α. One can show that the subspace, H α are mutually orthogonal. Theorem 5.3. H fermi = α Ar H α. (5.) In technical terms, irreducible representations of the symmetric group S n are in one-to-one correspondence with α A r and on the subspace, H α, the representation of S n is multiple to the irreducible one labeled by α. The spaces (5.7) and (5.8) correspond to the irreducible representations of S n, with the one-row and one-column Young diagrams, respectively. The remaining subspaces H α correspond to fermions with higher spins. Irreducible representations of S n can be connected, via Weyl s theory of dual pairs of groups, to irreducible representations of SU(2) carried by the spin space C (2r+)n and therefore determine the total spin of corresponding wave functions. To summarize, for identical particles the state space is not L 2 (R 3n ), but rather a subspace of it, defined by certain symmetry properties with respect to permutations of the particles. Homework 5.4. Find the ground state energy of H idealgas (in terms of the oneparticle energies) for (a) identical fermions of the spin 3/2 on the subspace corresponding to Young diagram with with the columns of the lengths 4, 4, 2, ; (b) 7 identical fermions of the spin /2 on the subspace corresponding to Young diagram with with the columns of the lengths 0, Multi-electron Atoms. Consider an atom consisting of a single nucleus of charge Ze and N = Z electrons. Assuming for simplicity that the nucleus is