Supplementary information I Hilbert Space, Dirac Notation, and Matrix Mechanics
Properties of Vector Spaces Unit vectors ~xi form a basis which spans the space and which are orthonormal ( if i = j ~xi ~xj = ij = 0 otherwise Column vectors ~z = zi = z ~x + z2 ~x2 + + zn ~xn 0 z B z2 C B C =B. C @.. A zn v v2 Row vectors ~v = hv = Inner product can be defined... vn ~v ~z = hv zi = v z + v2 z2 + + vn zn 2
Quantum States as Vectors: Dirac Notation The mathematical structure of QM is based on vector spaces. In QM, states correspond to vectors. For each state, we associate a ket i and bra h with it (analogous to column and row vectors). We can define the inner product between di erent states and as h i. In the case where the states can be Rrepresented as functions of position (x) and (x), h i = (x) (x)dx. Note that if i and i are distinct, h i may be complex. 3
Quantum States as Vectors Bras and kets obey the following rules (where a is a scalar): h a i = a h i ha i = a h i h i = h i h + = h + h and + i = i + i Therefore h + 2 + 2i =h i +h 2i +h 2 i +h 2 2i 4
Hilbert Spaces The vector spaces of interest in QM are Hilbert spaces. A Hilbert space H is a linear vector space whose elements are functions or vectors i with a positive-definite scalar product (i.e., h i > 0 and is finite). The dimensionality N of the Hilbert space is the number of linearly independent vectors/states needed to span it (may be finite or infinite). Properties: 2 3 4 Linearity: if i and i are elements of H, so is a + b. Inner product: h i exists and h i = h i. Every element i has a norm/length k k such that h i = k k2. Completeness: every Cauchy series of functions in H converges to an element in H. 5
Basis States of Hilbert Spaces For a Hilbert space of dimension N, we can choose N orthonormal states (basis states) i, 2 i, N i which span the space, such that any state i in H can be decomposed into i= N X n= bn ni where bn are scalars. This definition holds in infinite dimensions N! provided N P bn 2 <. n= 6
Orthonormality of Basis Orthonormal basis: a set of basis vectors which are pairwise orthogonal. Two vectors/states are orthogonal if h m 6= n. i of unit norm, m ni Normalization: vector/state is normalized if h = 0 for m mi =. We postulate that the set of eigenstates of any observable is orthonormal and hence is a possible basis for the Hilbert space (provable for finite dimensions, axiomatic for infinite dimensions). 7
Basis Expansion and the Born Rule Suppose we express an arbitrary state i in terms of the orthonormalized eigenstates i of an observable A such that X i= bn n i n Then let us examine the inner product X X h i i = bn h i n i = bn n i,n = bi n The Born rule ( generalized statistical interpretation ) states that a measurement of A performed on i has probability h i i 2 of returning the eigenvalue ai, leaving the system in the eigenstate i i of A. Therefore the coefficients bn are the probability amplitudes of observing i upon measurement! n 8
Dual Space We define a dual space composed of bra states h adjoint to the original ket Hilbert space. This allows us to perform operations between vectors/states. The analogy between column and row vectors If 0 is useful. b B b2 C C we write the coefficients of state as i = B @... A bn then the corresponding bra is a row vector h = b b 2... b N. If h is the adjoint of i, the adjoint of a i (where a is a scalar) is given by a i = a i () ha = h a 9
Operators in Dirac notation Operators transform one state (ket vector) into another state in H A i = i We can define an adjoint operator A that acts on the adjoint vector (bra): h A = h For linear operators and scalar a: A a i = aa i (A + B ) i = A i + B i A ( i + i) = A i + A i 0
Projection and Identity Operators The projection operator P i picks out ( projects ) a particular component of a state vector P i = i i h i. P For any i = bn n i, n P P P i i = i i h i bn n i = i i bn h i n i = n n P i i bn in = bi i i n The identity operator Iˆ always returns the state it is applied to Iˆ i = i and h Iˆ = h P P Note that Iˆ = ni h n = P n (a very useful result) n n If the eigenvalues indexed by n range over a continuous set of values, the summation becomes an integration R Iˆ = n i h n dn
An Infinite-Dimensional Hilbert Space: L2 The Hilbert space L2 (a, b) is the set of all square-integrable functions f (x) on the interval [a, b], i.e., f (x) such that Rb a f (x)f (x)dx < The inner product in L2 (a, b) is defined as h i= Z b (x) (x)dx a Examples include L2 (0, a) for the infinite square well and L2 (, ) for the free particle. Note the infinite dimensionality of the Hilbert spaces (evidenced by the infinite number of energy eigenfunctions, which comprise possible bases for these spaces). 2
Interregnum: The Dirac Delta Function We define the Dirac delta function via ( for x = 0 (x) 0 otherwise Z (x)dx = Note the units of (x) are x. The Dirac delta function picks out the origin. Operationally, for any function f (x), f (x) (x a) = f (a) (x because for any integral Z Z f (x) (x a)dx = f (a) (x a) a)dx = f (a) 3
The Dirac Delta Function cont ed The Dirac delta function is not a traditional mathematical function, though it can be seen as the limit of a variety of sharply peaked well-defined functions, for instance x sin!0 x (x) = lim Note that if we take f (x) = x, we obtain x (x x0 ) = x0 (x x0 ) This looks like an eigenvalue equation for the operator x. The implication is that the eigenfunctions of the position operator are Dirac delta functions centered at particular eigenvalues x0. 4
The Dirac Delta Function cont ed f (x) = p 2 If f (x) = (x Z F (k)e ikx dk, F (k) = p 2 x0 ), then F (k) = (x x0 ) = 2 exp( ikx0 ) p. 2 Z eik(x x0 ) Z f (x)e ikx dx This implies dk The Fourier transform of a delta function has constant magnitude F (k) 2 = 2. A function completely localized in position space is completely delocalized in momentum space, and vice versa. 5
Basis States for Position/Momentum Space For the Hilbert space L2, we can choose real space x or momentum space p as our basis. For x, the eigenfunctions/eigenvalues are (x x0 ) for all x0 in the range of L2. For p, we can define the eigenfunctions p exp(ipx/~) with definite momentum p = ~k. 2 ~ Relationships between basis states: Z hp p2 i = ei(p2 p )x/~ dx = (p p2 ) 2 ~ hx x2 i = Z (x hx p i = p 2 ~ Z x ) (x x2 )dx = (x (x x )eip x/~ dx = p x2 ) exp(ip x /~) 2 ~ 6
Position and Momentum Space, cont ed Note that though the position and momentum eigenfunctions are not square-integrable (and hence technically outside the Hilbert space), they are orthonormal in the Dirac sense. This is generally the case for operators whose eigenvalues are continuous. However, it is still extremely useful to use these states as basis functions, so we can write a general state as Z i = dx xi hx i Z = dp pi hp i Note that because we are dealing with a continuous rather than discrete range of eigenvalues, we integrate rather than sum over all possible eigenvalues. 7
Position and Momentum Space cont ed i= Z dx xi hx i = Z dp pi hp i hx i = (x), i.e., the value of the wave function at position x is simply the projection of the state i onto the position eigenstate xi. This makes our original interpretation of (x) 2 as the probability of measurement of x consistent with and subordinate to the Born rule that the probability = hx i 2. Likewise we can interpret (p) = hp i as the momentum space wave function, i.e., probability amplitude for measurement of p. Complete information about the state can be obtained from (p) or (x); they are simply projections of i onto di erent bases. R Consistency of inner product definitions: using Iˆ = xi hx dx Z Z Z h i = h ( xi hx dx) i = h xi hx i dx = (x) (x)dx 8
Position and Momentum - Fourier Transforms Conversion between Similarly (x) and (p): Z (p) = hp i = hp xi hx i dx Z dx = e ipx/~ (x) p 2 ~ R (x) = eipx/~ (p) pdp. 2 ~ The conversion between position and momentum space is mathematically a Fourier transform because hx pi = p2 ~ exp(ipx/~). 9
Discrete vs. Continuous Spectra Let A an i = an an i (where the eigenvalues an are discrete) and B bn i = bn bn i (where the eigenvalues bn are continuous) Discrete Continuous ham an i = mn P m am i ham = P i = m am i ham i P 2 m ham i = hbm bn i = (bm bn ) R dbm bm i hbm = R i = dbm bm i hbm i R dbm hbm i 2 = ham A an i = an mn hbm B bn i = bn (bm where mn is the Kronecker delta function and (bm Dirac delta function. bn ) bn ) is the 20
Operators - Matrix Formulation We have seen that quantum states can be represented as vectors in a vector space of dimension N. A linear operator can be represented by an N N matrix that operates on bra and ket vectors. It is completely defined by its actions on the basis vectors. If we know A n i = 0n i for each n we can write X X X A i = A bn n i = bn A n i = bn 0n i = 0 i n n n Here the set of states n i is some orthonormal basis (not necessarily eigenstates of A ), while the set 0n i is generally not orthonormal. 2
Operators - Matrix Formulation cont ed Let us express 0 i = P m b0m mi by projecting it back onto the original basis. We can find each amplitude b0m of the new state! X 0 0 0 bm = h m i = h m bn n i n =h = m X n ) b0m = X h X n bn A ni! m A n i bn Amn bn n We have N 2 scalars Amn = h m A n i which depend only on the basis states { m i}, not i or 0 i. 22
Operators - Matrix Formulation cont ed We can therefore write 0 i = A i as a matrix equation. 0 0 h b0 B b0 C B h B 2C B 0i = B. C = B @.. A @ b0n A i 2 A i... h N A h h A 2 i 2 A 2 i... i h A N 2i 0 b C B C B b2 C C B.. C.. C A @.. A bn The scalars Amn = h m A n i are the matrix elements of A in the basis set i. P In Dirac notation, we can write A = Amn m i h n. m,n 23
Adjoint Operators For scalars we have a i = a i! ha = h a For adjoint operators: A i = A i! ha = h A If A i = 0 i! h A = h 0 If Amn = h m A n i then A mn = h m A = ha ni m ni =h n A m i =h n A m i Therefore A mn = A nm. Adjoints are the transpose conjugate of the operator matrix. a b a c A=!A = c d b d 24
Adjoints cont ed It can be shown that (A B ) = B A. In general, to find the adjoint of bras, kets, operators, or scalars: Reverse order of all factors A! A i! h a! a. Note that in an quantity like h A i, A can operate to the right (as A on the ket) or to the left (as A on the bra). The same result will be obtained in either case. 25
Hermitian Operators and Observables Hermitian operators A have the property that A = A : This implies h A i = ha i. In matrix notation, Amn = A nm. will be of the form 2 A A2 6A A22 6 2 A = 6 A A 23 4 3...... Any Hermitian matrix 3 A3 A23 7 7 A33 7 5... All eigenvalues of Hermitian operators are real. Therefore, (by postulate), all operators for physical observables are Hermitian (because measured quantities are real numbers). An operator is anti-hermitian if A = A. 26
Hermitian Operators: Reality Proof of the reality of eigenvalues of Hermitian operators: If n i and an are the sets of eigenvectors and eigenvalues of A, then A n i = an n i and h n A = h n a n. Then h n A n i = an h n n i = a n h n n i depending on whether A operates to the left or right. Therefore an = a n, so an must be real. 27
Hermitian Operators: Orthogonality Proof of the orthogonality of eigenvectors of Hermitian operators: h m A n i = an h m n i (operating to the right) and h m A n i = a m h m n i (operating to the left) Therefore an h m n i = a m h m n i If an 6= am then h m n i = 0, i.e., h m n i = mn. Nondegenerate eigenstates of Hermitian operators are necessarily orthogonal. If di erent eigenstates are degenerate (i.e., share the same eigenvalue), we can always construct linear combinations of them which are orthonormal (for example using the Gram-Schmidt procedure). 28
Commutators Commutators between two operators are defined as [A, B ] = A B B A. Clearly [A, B ] = [B, A ]. Two operators A, B commute (or are compatible) if [A, B ] = 0. To figure out commutation relations, apply the operators on a test function and look at the end result (sans test function). Example: the canonical commutation relation [x, p ] = i~. 29
The Uncertainty Principle (again) Consider two operators A, B which don t commute, and whose commutator [A, B ] = ic where A, B, C are Hermitian. From linear algebra, the Schwarz inequality for any two vectors i, i states that h i h i h i 2 Take i = (A hai) i and i = (B hbi) i where hai, hbi are expectation values over i and must be real. 2 h i = h (A hai)2 i = A 2 h i = h (B hbi)2 i = B For any complex z 2 2 2 2 2 z = [Re(z)] + [Im(z)] [Im(z)] = (z z ) 2i 30
The Uncertainty Principle, cont ed For z = h i this implies 2 2 A B h i = h (A hai)(b = ha B i hai hbi = h A B i [h i 2i hbi) i hbi h A i By similar reasoning h i = hb A i h i h i = ha B i ) 2 2 A B h i] 2 hai h B i + hai hbi h i hai hbi hb A i = h[a, B ]i = ic 2 h[a, B ]i = hci 2 2i 3
The Uncertainty Principle, cont ed Example: [x, p ] = i~ 2 2 x p ) x 2 ~ 2 ~ p 2 Note that if two operators commute (are compatible), it is possible that the same state will be an eigenfunction of both operators. Then the two corresponding observables can be simultaneously specified for that state. The eigenvalues of the observables are good quantum numbers of the state. 32
Some Equivalences Wave Functions b (~ r) R A d~ r = Vectors 0 b B b2 C B C B. C @.. A i bn b N (x ) B (x2 )C @ A... 0 b 02 b B b2 C B C b b 2 b N B. C @.. A b 0 0 N b c b C Bc C B B 2C B 2C A B. C=B. C @.. A @.. A bn Dirac Notation h hr i h i A i = i cn 33
Some Equivalences cont ed Wave Functions hai = A n (x) R A = an Vectors b b 2 a B0 B B @ 0 a2 dr n (x) 0.. 0 R r m n d~ = mn 0. 0 0 b b C B B 2C b N A B. C @.. A bn 0 0 C C C in n basis A Dirac Notation h A i A ni = an ni an 0 0 B C B C 0 B.C = @.. A mn h m ni = mn 0 34
Unitary Operators ˆ Unitary operators U are such that U U = U U = I. This implies U = U. Note that unitary operators are not in general Hermitian and vice versa. The application of an unitary operator on a state leaves the norm unchanged. Unitary operators are useful because they transform between di erent choices of basis states in H. 35
H as the Generator of Time Evolution P P Let (t0 )i = i i i h i (t0 )i = i bi (t0 ) i i where { i } are eigenstates of H with corresponding energies {~!i }. From the time-dependent SE, the state at some later time t is! ih (t t0 ) (t)i = U (t t0 ) (t0 )i = exp (t0 )i ~! ih (t t0 ) X = exp bi (t0 ) i i ~ i X = bi (t0 )e i!i (t t0 ) i i i Here we define U (t t0 ) exp ih (t t0 ) ~! = i(t t0 ) ~ H + 2! i(t t0 ) ~ 2 The Hamiltonian generates the time evolution of states. U (t unitary. H 2 +... t0 ) is 36
Momentum as the Generator of Spatial Translation Define the translation operator T (a) such that T (a)f (x) = f (x + a). (T translates a state by distance a.) ip a d d d2 2 T (a) = exp = exp a =+ a+ a... ~ dx dx 2! dx2 df (x) d2 f (x) 2 T (a)f (x) = f (x) + a+ a = f (x + a) dx 2! dx2 Note the similarity between T! (a) and the time evolution 0 ih t operator U (t0 ) = exp. Both are unitary operators. ~ We say momentum generates spatial translations. 37
Ehrenfest Theorem What is the time dependence of expectation values? Z d hai d d = h (t) A (t)i = (x, t)a (x, t) dx dt dt dt * + @ @ A @ = h (t) A (t)i + + h (t) A (t)i @t @t @t * + @ A = hh A i h A H i + i~ @t * + i @ A = h[h, A ]i + ~ @t If A has no explicit time dependence, then dhai dt = i ~ D E [H, A ]. 38
Ehrenfest Theorem, cont ed 2 p Consider a Hamiltonian H = 2m + V (x). d hpi i i d = h[h, p ]i = V (x), i~ dt ~ ~ dx Z Z @ (x, t) @ = (x, t)v (x) dx (x, t) (V (x) (x, t))dx @x @x Z dv (x) = (x, t) (x, t)dx dx Therefore d hpi = dt dv (x) dx The quantum expectation values reproduce classical mechanics. 39
Symmetry and Conservation Laws Any generator that leaves the Hamiltonian of a system invariant corresponds to a conserved quantity of that system (Noether s theorem). Examples: Translational invariance implies momentum conservation: [H, p ] = 0 for free particle (H = p 2 /2m) implies dhpi dt = 0. Temporal invariance implies energy conservation: [H, H ] = 0 for time-independent H implies dhei dt = 0. Rotational invariance implies angular momentum ~ = 0 for angular momentum L ~ implies conservation: [H, J] ~ dhli dt = 0. 40