Quantum Mechanics in a Nutshell


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1 Quantum Mechanics in a Nutshell by Vitaly Vanchurin 1 November 13, If you have comments or error corrections, please send them to the LYX Documentation mailing list:
2 Contents 01 Vector Spaces 1 02 Inner Product 2 03 Outer Product 3 04 Linear Operators 4 05 Adjoint Operators 4 06 Classical Physics 6 07 Quantum Physics 7 08 Schrodinger Picture 8 09 Heisenberg Picture Harmonic Oscillator Ladder Operators 12 i
3 CONTENTS 1 01 Vector Spaces For our purpose the most relevant vector space is a finite (or countable) dimensional space of vectors with complex components represented in the column matrix notation as ψ ψ 1 ψ n where ψ i C and ψ C n or C These are the ketvectors By definition the vector spaces comes with: 1 addition operation ψ 1 ψ n + ϕ 1 ϕ n = ψ 1 + ϕ 1 ψ n + ϕ n (1), (2) 2 multiplication by a scalar, that is, a complex number (or cnumber), ψ 1 z ψ 1 z = (3) ψ n z ψ n 3 and a zero vector, not to be confused with a vacuum state 0 Spanning set in a vector space is a collection of vectors ϕ 1, ϕ 2 ϕ n such that an arbitrary vector can be written as their liner combination, (4) ψ = a i ϕ i (5)
4 CONTENTS 2 02 Inner Product An inner product (, ) is a function which takes two vectors ψ and ϕ to produce a single complex number denoted by ( ψ, ϕ ) with following properties: 1 (, ) is linear in the second argument 2 ( ψ, ϕ ) = ( ϕ, ψ ) 3 ( ψ, ψ ) 0 with equality only iff ψ = 0 For example the vector space C n has an inner product defined by ψ 1 ψ n, ϕ 1 ϕ n ψi ϕ i (6) The dual vector ψ is a linear function from a vector space to complex numbers defined using the inner product, ψ ( ϕ ) ψ ϕ ( ψ, ϕ ) (7) In matrix representation it is a raw vector (also known as a dual vector, a covector or a bravectors), ψ (ψ 1, ψ n ) (8) when the column vector is known as a ketvector In a quantum mechanics a finite dimensional vector space with an inner product is called a Hilbert space A number of definitions follow Two vectors ψ and ϕ are orthogonal if ψ ϕ = 0 Norm of a vector ψ is ψ ψ ψ is a unit vector if ψ ψ = 1 a set of vectors i is orthonormal if i i = 1 and i j = 0 for i j
5 CONTENTS 3 With an orthonormal basis an inner product can be written in a matrix representation as ϕ 1 ψ ϕ = ψ i i, ϕ j j = ψi ϕ j i j = ψi ϕ j = (ψ1,, ψn) j=1 j=1 (9) ϕ n 03 Outer Product Another convenient representation of linear operators is an outer product representation defined by, ψ ϕ ( ϕ ) ψ ϕ ϕ = ϕ ϕ ψ (10) For example an identity operator can be represented as a sum of outer product of an orthonormal basis, ie Î = i i (11) This is known as a completeness relation which can be used to obtain an outer product representations of operators, n Â = ÎÂÎ = i Â j i j (12) j=1 Eigenvectors i and their respective eigenvalues λ i of a linear operator Â are defined by Â i = λ i i (13) In a matrix representation the eigenvalues can be determined from a characteristic equation det ( Â λî) = 0 (14) Diagonalizable representation of an operator (also known as a orthonormal decomposition) is given by where the eigenvectors i form an orthonormal set Â = λ i i i (15)
6 CONTENTS 4 04 Linear Operators Linear operator on a vector space is a function which is linear in its inputs, ( n ) A a i ϕ i = a i A( ϕ i ) (16) (To emphasize the difference from cnumber or complex numbers the operators are sometimes called the qnumbers or quantum numbers) The two simplest operators are the identity operator Î, defined by Î ψ = ψ for all ψ zero operator ˆ0, defined by ˆ0 ψ = 0 for all ψ Similarly to vectors, operators A : C n C n can be represented in terms of matrices In matrix representation the linearity condition (16) can be rewritten as, Â a i ϕ i = a i Â ϕ i, (17) where Â is an n n matrix and ϕ i s are the n 1 column vectors The entries A ij of the matrix Â can be obtained for a given set of basis vectors ϕ 1, ϕ 2 ϕ n from Â ϕ j = A ij ϕ i (18) i An important example of operators on C 2 are the Pauli matrices, σ 0 σ 1 σ 2 σ 3 ( ) ( ) ( ) 0 i i 0 ( ) 1 0 (19) Adjoint Operators For every linear operator Â on a Hilbert space there exist an adjoint (or Hermitian conjugate) operator defined as ( ψ, Â ϕ ) (Â ψ, ϕ ) (20)
7 CONTENTS 5 It follows that (AB) = B A (21) and (Â ψ ) = ψ Â, (22) where ψ ψ (23) In a matrix representation an adjoint operator is defined as Â ( Â ) T (24) where () is a complex conjugation and () T is a transpose operation Some useful definitions: Â is a positive definite operator if ( ψ, Â ψ ) is a positive real number for all ψ 0 Â is a selfadjoint (or Hermitian) operator if Â = definite operator is Hermitian Â Any positive Â is a normal operator if ÂÂ = Â Â For example, any Hermitian operator is normal ˆP is a projection operator if ˆP = k i i, where i is an orthonormal basis and k n Û is a unitary operator if Û Û = Î Any pair of orthonormal basis ψ i and ϕ i can be used to define unitary operators, ie Û = n ψ i ϕ i Some useful results: Unitary operators preserve the inner product, (Û ψ, Û ϕ ) = ψ Û Û ϕ = ψ Î ϕ = ψ ϕ (25)
8 CONTENTS 6 06 Classical Physics Classical physics is based on the two postulates: 1 State space postulate: Any closed system is associated with even dimensional space called the phase space The state is described by a single point (or vector) in the phase space The state is specified by N position (usually denoted by q i s) and N momentum (usually denoted by p i s) coordinates: (q 1, q 2,, q N, p 1, p 2,, p N ) (26) What is the dimensionality of the phase space of a simple harmonic oscillator? What is the dimensionality of the phase space of a single particle in a box? How many real numbers needed to specify a state of the harmonic oscillator and how many needed to specify a single particle? 2 Evolution postulate: Evolution of any closed system is described by function on the phase space The function is called Hamiltonian and denoted by H(q 1, q 2,, q N, p 1, p 2,, p N ) and the evolution is described by the following equations ṗ i = H q i q i = H p i (27) For a simple harmonic oscillator the Hamiltonian is and the corresponding equations of motion are H(q, p) = p2 2m + kq2 2 (28) ṗ = kq q = p m (29) or equivalently q = ṗ m = kq m (30)
9 CONTENTS 7 07 Quantum Physics In contrast the quantum physics is based on three postulates We will first state the postulates and then introduce the necessary mathematical formalism that goes with it 1 State space postulate: Any closed system is associated with a Hilbert space The state of the system is described by a single point (or ketvector) in the Hilbert space with unit length where ψ is a bravector ψ (31) ψ ψ ψ ψ = 1 (32) 2 Evolution postulate: Evolution of any closed system is described by a unitary operator, ie where ψ(t 2 ) = Û(t 2 t 1 ) ψ(t 1 ) (33) Û(t 2 t 1 ) e i (t 2 t 1 )Ĥ (34) and Ĥ is a Hermitian operator known as Hamiltonian operator 3 Measurement postulate: A measurement is described by a collection of measurement operators { ˆM m } with probability of an outcome m given by p(m) = ψ ˆM m ˆM m ψ (35) where ˆM m is the Hermitian conjugate of ˆMm and the state after measurement ˆM m ψ p(m) (36) So to understand the quantum mechanics we have to understand what is a Hilbert space, ket and bravectors, unitary and Hermitian operators, etc The branch of mathematics which studies vector spaces and linear operators on those spaces is linear algebra
10 CONTENTS 8 08 Schrodinger Picture One can show that all normal operators have a diagonal representation, Â = λ i i i (37) Therefore all positive definite operators as well as Hermitian operators have diagonal representations Then one can define a function of Hermitian operators as n f(â) f(λ i ) i i (38) for an arbitrary function f This can be applied to Eq (33) since the Hamiltonian operator Ĥ must be Hermitian and thus has a spectral decomposition Ĥ = E n ψ n ψ n (39) where E n and ψ n are the eigenvalues and corresponding eigenstates of Hamiltonian operators, ie Ĥ ψ n = E n ψ n (40) The energy eigenstate ψ n with the lowest energy eigenvalue E n is called the ground state Moreover, by expanding both sides of Eq (33) we get ψ(t 1 ) + (t 2 t 1 ) d ( dt ψ(t 1) = 1 (t 2 t 1 ) i ) ψ(t 1 ) (41) Ĥ which is the famous timedependent Schrodinger equation i d ψ(t) = Ĥ ψ(t) (42) dt in contrast to the timeindependent Schrodinger equation (40) Using (33), (34), (38) and (39) n can find the most general solution of the timedependent Schrodinger equation ϕ (t) = n e ient [ ψ n ϕ (0)] ψ n (43) which is expressed as a linear sum over solutions of the timeindependent Schrodinger equation Note that each solution of the timeindependent Schrodinger equation gives rise to a simple solution of the timedependent Schrodinger equation ϕ (t) = e ient ψ n (44)
11 CONTENTS 9 09 Heisenberg Picture A framework where the state vectors evolves with time, but the operators remain constant is a Schrodinger picture There is also a Heisenberg picture where the operators change with time, and the state vectors remain constant Consider a time independent Hermitian operator ÂS in the Schrodinger picture then ÂS (t) = ψ S (t) ÂS ψ S (t) (45) The time evolution is described by a Schrodinger equation whose solution is From (45) and (46) we obtain where ψ S (t) = e i Ĥt ψ S (0) (46) ÂS (t) = ψ S (0) e i Ĥt Â S e i Ĥt ψ S (0) = ψ H ÂH(t) ψ H = ÂH(t) (47) Â H (t) e i Ĥt Â S e i Ĥt (48) ψ H ψ S (0) (49) It is easy to calculate the time evolution of operators in Heisenberg picture, dâh(t) dt = d dt e i Ĥt Â S e i Ĥt + e i Ĥt Â S d = i Ĥe i Ĥt Â S e i Ĥt e i Ĥt Â S i = i (ĤÂ H (t) ÂH(t)Ĥ) i dt e Ĥt Ĥe i Ĥt = i where the commutator is defined as [Ĥ, Â H (t) ] (50) [Â, ˆB] Â ˆB ˆBÂ (51) Expression (50) is known as the Heisenberg equations It reduces to the Hamiltonian equations of motion in classical mechanics with commutators replaced by Poisson brackets, i [, ] {, } q p p q (52)
12 CONTENTS 10 ie dp dt dq dt = {p, H} = H q = {q, H} = H p (53) (54) Moreover, the Poisson bracket between conjugate variables is replaced by a commutation relation {q, p} = 1 (55) i [ˆq, ˆp] = 1 (56) 010 Harmonic Oscillator The state of classical harmonic oscillator is described by position q(t) and momentum p(t) coordinate satisfying equations of motion (53) and (54) with Hamiltonian H = p2 2m + kq2 2 (57) The corresponding second order differential equation of motion is q + k q = 0 (58) m Thus, for classical harmonic oscillator the lowest energy solution is given by q(t) = q(t) = 0 To find a solution corresponding to the ground state we work in the position basis where q are the position eigenvectors, ie and define the wave function as ˆq q = q q (59) Ψ(q) q ψ (60) To quantize the harmonic oscillator we first defined a vector space of square integrable complex valued functions, ie f(q) 2 dq < (61)
13 CONTENTS 11 with an inner product given by, g f g(q) f(q)dq (62) It is easy to show that the space of squareintegrable complex valued functions is a Hilbert space Then we can replace the classical coordinated by quantum operators satisfying (56), eg q ˆq q (63) p ˆp i q (64) and rewrite the timeindependent Schrodinger equation as ( with a ground state 2 ( ˆp 2 2m + kˆq2 2 2 ) 2m q + 2 kq2 Ψ(q) exp corresponding to the zeropoint energy ( ) ψ = E ψ (65) Ψ(q) = EΨ(q) (66) ) km 2 q2 (67) ( ) ( ) 2 km E 0 = 2 = k 2m 2 2 m (68) where ω k/m is the angular frequency Then the solution of a timedependent Schrodinger equation is Ψ(q, t) = ( km π ) 1 4 exp ( i E 0 t ) km 2 q2 where the normalization is determined from Ψ(q) Ψ(q) = 1
14 CONTENTS 12 Then, for example, a set of measuring operators {M q = q q } describes probabilities (given by (35)) for detecting the particle in position q, p(q) = ψ M q ψ = ψ q q ψ = q ψ q ψ = Ψ(q) Ψ(q) ( ) km 1 ( ) 2 km = exp π 4 q2 This is the famous Born rule which is the additional third postulate of quantum mechanics which has no analog in classical physics It is a subject of ongoing debated whether the postulate can be derived from the first two (ie the state space and evolution postulates) 011 Ladder Operators It is convenient to define lowering â mω 2 ( ˆx + i ) mω ˆp and raising operator ( mω â ˆx i ) 2 mω ˆp which are useful for expressing other operators (69) (70) ˆx = (â + â ) (71) 2mω mω ˆp = i (â â ) (72) 2 ( Ĥ = â â + 1 ) ( ω = ââ 1 ) ω (73) 2 2 with following commutation relations, [â, â ] = 1 (74) [Ĥ, â ] = â ω (75) [Ĥ, â] = â ω (76)
15 CONTENTS 13 If we now denote the energy eigenstates ψ n with eigenvalues E n, ie then the commutation relations imply Ĥ ψ n = E n ψ n, (77) Ĥ â ψ n = (E n + ω)â ψ n (78) Ĥ â ψ n = (E n ω)â ψ n (79) This means that â ψ n and â ψ n are also eigenstates of the Hamiltonian with energies E n + ω and E n ω That is why operators â and â are called the lowering and rising operators between energy eigenstates However the eigenvalues cannot be arbitrarily small, ( E n = ψ n Ĥ ψ n = ψ n â â + 1 ) ω ψ n = (â ψ n ) â ψ n ω 1 2 ω (80) and thus, the ground state is defined by, â ψ 0 = 0 (81) in agreement with (67) Additional application of the lowering operator would not produce any new eigenstates, but the raising operator can be used to produce all of the excited states, ψ n = 1 (â ) n ψ0 (82) n! with energies, E n = ( n + 1 ) ω (83) 2
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