Some Linear Algebra and Elementary Quantum Mechanics

Representation theory and quantum mechanics tutorial Some linear algebra and elementary quantum mechanics

Justin Campbell July 25, 2017

1 Hilbert spaces

1.1 Recall that a real inner product space is a vector space V over R equipped with a bilinear form

h , i : V × V −→ R satisfying (i) (symmetric) we have hv, wi = hw, vi for all v, w ∈ V , (ii) (positive-deﬁnite) we have hv, vi > 0 for all v ∈ V such that v 6= 0. In a real inner product space V one can make sense of the length of a vector v ∈ V :

|v| := phv, vi,

as well as the angle θ between two vectors v, w ∈ V , using the formula hv, wi cos θ = . |v||w|

In particular, two vectors v, w ∈ V are orthogonal if and only if hv, wi = 0. For example, the Euclidean space Rn with the dot product is an inner product space. The Gram-Schmidt algorithm shows that any n-dimensional real inner product space V admits an orthonormal basis, meaning a basis consisting of pairwise orthogonal vectors of length one. The resulting isomorphism V →˜ Rn sends the inner product in V to the dot product in Rn. But many interesting inner product spaces are inﬁnite- dimensional.

Exercise 1.1.1. Let C∞(S1) be the space of smooth R-valued functions on the circle, which we can view as smooth functions f : R → R which are periodic of period, say, 2π. Show that C∞(S1) is an inner product space under the pairing Z Z 2π hf, gi := f(θ)g(θ) dθ = f(θ)g(θ) dθ. S1 0

If we work with vector spaces over the complex numbers C, as we must to do quantum mechanics, the positive-deﬁniteness condition forces us to abandon bilinear forms. Instead we use Hermitian, i.e. conjugate- symmetric, forms.

Deﬁnition 1.1.2. A Hilbert space or complete complex inner product space is a vector space H over C equipped with a form h , i : H × H −→ C satisfying

(i) the map v 7→ hv, wi is linear for any w ∈ H ,

1 (ii) (Hermitian) we have hv, wi = hw, vi for all v, w ∈ H , (iii) (positive-deﬁnite) we have hv, vi > 0 for all v ∈ H such that v 6= 0. (iv) (complete) any Cauchy sequence in H converges. Here the notions of Cauchy and convergent sequence are deﬁned using the norm

|v| := phv, vi,

which in turn gives rise to a metric. The “angle” θ between two vectors v, w in any complex inner product space can be defined so that Re hv, wi cos θ = . |v||w| In particular, they are orthogonal if and only if hv, wi = 0. Remark 1.1.3. The axiom of completeness is automatically satisfied in the finite-dimensional case. Any complex inner product space V injects into a canonical Hilbert space Vˆ , called its completion. Exercise 1.1.4. Construct the completion Vˆ ⊃ V of an arbitrary complex inner product space V , including its unique Hermitian form extending the one on V (hint: imitate the construction of the real numbers using Cauchy sequences). Show that for any Hilbert space H which contains V as a dense subspace with the induced inner product, there is a unique isomorphism Vˆ →˜ H which preserves the Hermitian forms.

The basic example of a Hilbert space is the standard n-dimensional complex vector space Cn with the pairing X h(ai), (bi)i := aibi, i which is the Hermitian version of the dot product. As in the real case, the Gram-Schmidt algorithm shows that any finite-dimensional Hilbert space admits an orthonormal basis. One can define an infinite-dimensional complex inner product space C∞(S1, C) analogously to Exercise 1.1.1: a vector is a periodic function f : R → C of period 2π, and the inner product is defined by Z Z 2π hf, gi := f(θ)g(θ) dθ = f(θ)g(θ) dθ. S1 0 This space is not complete; its completion is the Hilbert space denoted by L2(S1).

1.2 Let H1 and H2 be Hilbert spaces and A : H1 → H2 a C-linear map. ∗ Deﬁnition 1.2.1. A (Hermitian) adjoint of A is a linear map A : H2 → H1 satisfying

∗ hAv, wiH2 = hv, A wiH1 for all v ∈ H1, w ∈ H2.

If H1 and H2 are ﬁnite-dimensional, then any map A : H1 → H2 admits a unique adjoint. Namely, n m one can choose orthonormal bases to identify H1 and H2 with C and C respectively, endowed with their Hermitian dot products. Then A is realized as a matrix, and its adjoint A∗ = A> is simply the conjugate transpose of A. This observation generalizes as follows. Theorem 1.2.2. A continuous linear map between Hilbert spaces admits a unique adjoint, which is then also continuous.

One proves this by reducing to the case of a continuous linear functional λ : H → C. Then the Riesz representation theorem guarantees that there exists a vector v ∈ H such that

λ(w) = hw, vi

2 for all w ∈ H . A continuous linear operator A : H → H is called self-adjoint or Hermitian if A = A∗, i.e. we have

hAv, wi = hv, Awi

for all v, w ∈ H . For example, let H = C with the pairing hz, wi = zw.A C-linear operator on C is multiplication by some z ∈ C, and z∗ = z. Thus z is self-adjoint if and only if z = z, i.e. z ∈ R. Remark 1.2.3. Many subtleties can occur in the case that H is infinite-dimensional, where one sometimes needs to consider discontinuous or partially-defined linear operators. Then this definition gives too large a class of self-adjoint operators, and one must also impose the condition that the domain of A∗ is equal to the domain of A. The key property of self-adjoint operators is expressed in the following theorem.

Theorem 1.2.4 (Spectral theorem for self-adjoint operators). If A : H → H is a self-adjoint operator, then the eigenvalues of A are real and H admits an orthonormal basis consisting of eigenvectors for A.

Proof. If v ∈ H is an eigenvector of A with eigenvalue λ, then we have

λhv, vi = hAv, vi = hv, Avi = λhv, vi.

Thus λ = λ, i.e. λ ∈ R. We give the rest of the proof under the assumption that H is finite-dimensional. The infinite-dimensional case is much deeper, and more subtle: for example, the definition of “basis” has to be modified. The theorem being obvious in the case that H = 0, we may proceed by induction, assuming that the theorem holds when dim H = d − 1. Suppose dim H = d and let v ∈ H be an eigenvector of A, which must exist because H is a finite-dimensional complex vector space. We can normalize so that |v| = 1. Let

W := {w ∈ H | hw, vi = 0}

be the subspace orthogonal to v. Then for any w ∈ W , we have

hAw, vi = hw, A∗vi = hw, Avi = λhw, vi = 0,

meaning A preserves the subspace W . Since dim W = d − 1, the inductive hypothesis implies that W admits an orthonormal basis consisting of eigenvectors for A. Adding v to this basis completes the proof.

1.3 A continuous linear operator A : H → H is called unitary if A∗ = A−1, or equivalently A is invertible and preserves the Hermitian inner product, i.e.

hAv, Awi = hv, wi

for v, w ∈ H .

Exercise 1.3.1. Consider the Hilbert space Cn with the Hermitian dot product. Show that a linear map A : Cn → Cn, i.e. an n × n complex matrix, is unitary if and only the columns of A form an orthonormal basis of Cn. More generally, a linear operator A : H → H on an n-dimensional Hilbert space H is unitary if and only if for any orthonormal basis v1, ··· , vn of H , the vectors Av1, ··· , Avn form an orthonormal basis. Theorem 1.2.4 can be reformulated as follows for matrices: if A ∈ Matn×n(C) is self-adjoint, then there exists an n × n unitary matrix U such that UAU −1 is a diagonal matrix with real entries. The n × n unitary matrices form a group

∗ ∗ Un := {A ∈ GLn(C) | A A = AA = I}

3 called the nth unitary group. This is a real algebraic group, since

Un ⊂ GLn(C) ⊂ GL2n(R). One can also consider the nth special unitary group

SUn := Un ∩ SLn(C) = {A ∈ Un | det A = 1}. The ﬁrst unitary group 1 U1 = S = {z ∈ C | |z| = 1} is the circle group. Moreover, SU1 = 1 is the trivial group.

Exercise 1.3.2. Prove that |det A| = 1 for a unitary matrix A ∈ Un, and that the image of

det × Un ⊂ GLn(C) −→ C is U1.

1.4 Similarly to the case of the orthogonal group, one computes that the Lie algebra

∗ un := {A ∈ gln(C) | A = −A}, of Un consists of skew-Hermitian matrices.

Exercise 1.4.1. Show that a skew-Hermitian matrix A ∈ un has purely imaginary diagonal entries, and in particular that tr A is purely imaginary.

Similarly, the Lie algebra of SUn is

sun := un ∩ sln(C) = {A ∈ un | tr A = 0}.

Let H be a Hilbert space. Observe that A 7→ iA deﬁnes a bijection

{self-adjoint operators on H }−→{˜ skew-Hermitian operators on H } with inverse A 7→ −iA. Thus self-adjoint and skew-Hermitian operators are essentially interchangeable. The advantage of using self-adjoint operators is that the spectral theorem applies, and in particular that they have real eigenvalues, which is important for physical applications. On the other hand, skew-Hermitian operators are closed under the commutator bracket, whereas self-adjoint operators are not: the commutator bracket of two self-adjoint operators is skew-Hermitian.

2 Axioms of quantum mechanics

2.1 We now present the fundamental principles of quantum mechanics. The ﬁrst of these describes the state space of a quantum-mechanical system, analogous to phase space in classical mechanics.

Axiom 1 (State space). Any quantum-mechanical system has an associated Hilbert space H . A state of the system corresponds to a nonzero vector in H , and two state vectors are physically indistinguishable if they are proportional (i.e. complex scalar multiples of each other).

Thus the “true” state space, whose points are physical states, is the projective space

× P(H ) := (H \{0})/C , but as we will see the dynamics of the system really take place in H . We will therefore, by a slight abuse of terminology, refer to H as the state space of the system.

4 Note that two normalized (i.e. length one) vectors in H correspond to the same physical state if and × only if they differ by a phase, meaning they are proportional via an element of U1 ⊂ C . The simplest nontrivial quantum system is a “two-state system” or “qubit,” whose state space H = C2 with the Hermitian dot product. Despite the name, this system has infinitely many possible states: the 1 nomenclature will become more clear after we discuss measurement. A basic example is a spin 2 particle such as an electron, fixed in some manner so that we can ignore its motion. Then its internal state is characterized by a quantity called “spin” which is, as we shall see, analogous to classical angular momentum. 2 1 1 For this system the standard basis vectors e1, e2 ∈ C are usually denoted by |+ 2 i and |− 2 i, so that a general state has the form 1 1 z|+ 2 i + w|− 2 i for some z, w ∈ C. If we are thinking of this system as a qubit then the basis vectors might be labelled |0i and |1i.

2.2 Recall that in classical mechanics, observable quantities are functions on phase space.

Axiom 2 (Observables). Observable quantities of a quantum-mechanical system with state space H correspond to self-adjoint operators on H . Equivalently, observables correspond to skew-Hermitian operators, since these are in bijection with self- adjoint ones. Consider a two-state system with H = C2. The space of self-adjoint 2 × 2 matrices is 4-dimensional, with the conventional basis consisting of the identity I together with the Pauli matrices

0 1 0 −i 1 0 σ = , σ = , σ = . 1 1 0 2 i 0 3 0 −1

1 1 1 1 In the case of a spin 2 particle, the operators 2 σ1, 2 σ2, and 2 σ3 correspond to the quantities of spin measured with respect to the x-, y-, and z-axes respectively, with the orientation determined by the right-hand rule. General linear combinations aσ1 + bσ2 + cσ3 for a, b, c ∈ R correspond to (scalar multiples of) spin measured with respect to other oriented lines. We point out that iσ1, iσ2, and iσ3 form a basis for the Lie algebra su2 of traceless skew-Hermitian matrices, which will be important when we explain the relationship of spin to rotational symmetry.

2.3 Next, we introduce Schrödinger’sequation, which governs the time evolution of quantum-mechanical states in time. It is analogous to Hamilton’s equation in classical mechanics. Axiom 3 (Dynamics). There is a distinguished self-adjoint operator H called the Hamiltonian, corresponding to the observable quantity of energy. The evolution of a state vector ψ ∈ H in time satisfies the Schrödingerequation dψ i = Hψ. ~ dt h −34 Here ~ = 2π , where h ≈ 1.05457 × 10 Joule-seconds is Planck’s constant. It is possible to choose units so that that ~ = 1, and in what follows we will often implicitly make this choice. d Remark 2.3.1. Similarly to the situation with Hamilton’s equation, one must interpret the operation dt suitably. Namely, if t 7→ ψ(t) is a physical trajectory in the state space H , then Schrödinger’sequation says that

dψ i~ = Hψ(0). dt t=0 Recall that for H ﬁnite-dimensional and A : H → H any operator, one has the matrix exponential

A A2 A3 e := I + A + 2! + 3! + ··· , which always converges to an invertible operator on H .

5 Exercise 2.3.2. Prove that for A a skew-Hermitian operator, the exponential eA is unitary. Deduce that e−itH is unitary for all t ∈ R using the self-adjointness of H. Prove that for any state vector ψ = ψ(0) ∈ H , the trajectory ψ(t) = e−itH ψ satisﬁes the Schr¨odingerequation.

1 In our example of a motionless spin 2 particle, the Hamiltonian operator is (up to a positive scalar factor)

H = B1σ1 + B2σ2 + B3σ3,

where B = (B1,B2,B3) is the magnetic ﬁeld at the location of the particle. Assume for simplicity that B1 = B2 = 0, i.e. the magnetic ﬁeld points in the z-direction, so that H = B3σ3. One can compute that

e−itB3 0 e−itH = e−itB3σ3 = . 0 eitB3

Suppose that at time t = 0 the particle is in the state

1 1 ψ(0) = z|+ 2 i + w|− 2 i where z, w ∈ C. It follows from Exercise 2.3.2 that at time t, the particle is in state

−itB3 1 itB3 1 ψ(t) = ze |+ 2 i + we |− 2 i.

1 1 For instance, if ψ(0) = |+ 2 i or |− 2 i, then ψ(t) diﬀers from ψ(0) only by a phase factor and therefore the particle stays in the same physical state for all time. One says that these are steady states. On the other hand, if both z and w are nonzero, then the particle’s spin changes over time, with the axis processing at a rate determined by strength B3 of the magnetic ﬁeld.

2.4 The Schr¨odingerequation is deterministic: as a diﬀerential equation, it uniquely determines a physical trajectory from initial conditions. But the predictions of quantum mechanics are probabilistic, and indeed we have said nothing so far about the results of measurements. This is necessary because we do not observe complex linear combinations or “superpositions” of states: some explanation is required to specify what happens when we try to measure a system in such a state.

Axiom 4 (Measurement). Let A be a self-adjoint operator on H , and use the spectral thorem to ﬁnd an orthonormal eigenbasis {ψi} with respect to A, where ψi has eigenvalue λi ∈ R. Suppose ψ ∈ H is a P normalized state and write ψ = i ziψi. If the observable quantity corresponding to A is measured, and the 2 system is in state ψ before the measurement, then probability that the measurement will yield λi is |zi| . At the time of measurement the system is in state ψi. We pause to record some consequences of the measurement axiom. First, when measuring an observable quantity corresponding to a self-adjoint operator, the measurement will yield one of the eigenvalues of A with probability 1. Second, if the state of the system before the measurement is an eigenvector ψi of A with eigenvalue λi, then the probability that the measurement will yield λi is 1. Third, the expected value (i.e. mean over many measurements) of the observable quantity when the system is in the normalized state P ψ = ziψi is X 2 hψ, Aψi = hAψ, ψi = λi|zi| . i

Note that this expected value need not be equal to λi for any i, even though every measurement yields λi for some i. Remark 2.4.1. There is something unsatisfactory about the measurement axiom as we have stated it. One reason is that it is too complicated, and one would hope that it could be derived from more fundamental principles. After all, measurement is not usually thought of as a fundamental phenomenon. Another reason to complain is that in general the state of the system apparently makes a discontinuous jump at the moment

6 of measurement, from a superposition of eigenstates to an eigenstate. This “wave function collapse” is objectionable because, for example, it would seem to violate the Schr¨odingerequation. Various attempts have been made to ﬁx this problem, from the Copenhagen interpretation (formulate quantum mechanics as we have and ignore the problem of measurement) to the many-worlds interpretation. Another approach using “decoherence” is currently popular, and plays an important role in the theory of quantum computation.

1 Returning to our previous example of a spin 2 particle in a magnetic ﬁeld B = (0, 0,B3), suppose that the particle is in the normalized state

1 1 ψ = ψ(0) = z|+ 2 i + w|− 2 i 1 and we measure the spin with respect to the z-axis. The corresponding observable is A = 2 σ3, for which 1 1 1 |+ 2 i and |− 2 i are eigenvectors with their eponymous eigenvalues. We will observe + 2 (spin “up”) with 2 1 2 probability |z| , and − 2 (spin “down”) with probability |w| . We showed above that the state of the particle at time t is −itB3 1 itB3 1 ψ(t) = ze |+ 2 i + we |− 2 i. Since |e−itB3 | = |eitB3 | = 1, the probabilities do not change with time. In particular the expected value of the spin is 1 1 2 1 2 2 1 hψ(t), 2 σ3ψ(t)i = 2 |z| − 2 |w| = |z| − 2 for all time. What if we measure the energy of the particle, so that A = H? The analysis is quite similar to the case 1 1 1 A = 2 σ3, since the two operators are proportional. Only the eigenstates |+ 2 i and |− 2 i have a well-deﬁned energy, namely B3 and −B3 respectively. For a normalized state

1 1 z|+ 2 i + w|− 2 i,

2 the probability that measuring the energy will yield B3 is |z| , and the probability that measuring the energy 2 2 1 will yield −B3 is |w| = 1 − |z| . Like the case A = 2 σ3, these probabilities are constant in time, as is the expected value of the energy 2 2 2 B3|z| − B3|w | = 2B3|z| − B3. 1 Finally, we consider the observable A = 2 σ1, which corresponds to measuring spin with respect to the 1 x-axis. Orthonormal eigenstates for 2 σ1 are given by

ψ := √1 (|+ 1 i + |− 1 i)) and ψ := √1 (|+ 1 i − |− 1 i), 1 2 2 2 2 2 2 2

1 1 1 1 with eigenvalues 2 and − 2 respectively. With ψ(0) = z|+ 2 i + w|− 2 i as above recall that

−itB3 1 itB3 1 ψ(t) = ze |+ 2 i + we |− 2 i. Since |+ 1 i = √1 (ψ + ψ ) and |− 1 i = √1 (ψ − ψ ), 2 2 1 2 2 2 1 2 we obtain ψ(t) = √1 (ze−itB3 + weitB3 )ψ + √1 (ze−itB3 − weitB3 )ψ . 2 1 2 2 For example, if ψ(0) = ψ , i.e. z = w = √1 , it follows that 1 2

ψ(t) = cos(B3t)ψ1 + sin(B3t)ψ2.

Thus the expected value of spin with respect to the x-axis at time t is

1 1 2 1 2 1 hψ(t), 2 σ1ψ(t)i = 2 cos (B3t) − 2 sin (B3t) = 2 cos(2B3t). That is, after many measurements, one ﬁnds that the average value of spin with respect to the x-axis oscillates sinusoidally in time, with a frequency proportional to the strength of the magnetic ﬁeld B3.