Chap 3. Formalism Hilbert Space Observables Eigenfunctions of a Hermitian Operator Generalized Statistical Interpretation The Uncertainty Principle Dirac Notation

Vector Space & Inner Product Vector space : Linear space closed under vector addition & scalar multiplication.  Inner product :    : V  V   satisfying Conjugate symmetric. with  Positive. Linear. which implies

Dual Space Dual space V* of a vector space V  Set of all linear maps V* can be associated with an inner product on V by setting V* is itself a vector space and isomorphic to V. Thus, the dual   |  V* of a vector |    V is defined as the linear mapping such that

Quantum state space is a Hilbert space. State of system : Wave function Observables : Operators ( -D ) Vectors Linear transformations Linearity  Hilbert space  Complete inner product space. ( Cauchy sequence always converges ) E.g., Set of all square-integrable functions over a domain  : exists with inner product : L2  L2   : Quantum state space is a Hilbert space.

Schwarz inequality : ( cos   1 ) [ Guarantees inner product is finite in Hilbert space. ] conjugate symmetric orthogonal  r component of vector f  completeness Read Prob 3.1, Do Prob 3.2

Hermitian Operators Determinate States 3.2. Observables Hermitian Operators Determinate States

3.2.1. Hermitian Operators Expectation value of Q : Outcomes of experiments are real :     Q is hermitian (self-adjoint) Observable are represented by hermitian operators. E.g. : Read Prob 3.3, 3.5 Do Prob 3.4

3.2.2. Determinate States Determinate state : A state  on which every measurement of Q gives the same value q. i.e.,  Determinate states are eigenstates. E.g., solutions to the Schrodinger eq. H  E  are determinate states of the total energy, as well as eigenfunctions of the hamiltonian. Spectrum of an operator  Set of all of its eigenvalues If two or more independent eigenfunctions share the same eigenvalue, the spectrum is degenerate.

Example 3.1. Let , where  is the polar coordinate in 2-D. Is Q hermitian ? Find its eigenfunctions and eigenvalues. Ans. Consider the Hilbert space of all functions Q is hermitian. Eigenequation has eigenfunctions   Spectrum of Q is the set of integers, & it’s non-degenerate.

3.3. Eigenfunctions of a Hermitian Operator Discrete Spectra Continuous Spectra

Phys : Determinate states of observables. Math : Eigenfunctions of hermitian operators. Discrete spectrum : n  L2 normalizable & physically realizable. Continuous spectrum : k not normalizable & not physically realizable. Can be used to form wave packets. Examples: Purely discrete spectrum : Harmonic oscillator. Purely continuous spectrum : Free particle. Mixed spectrum : Finite square well.

3.3.1. Discrete Spectra Theorem 1 : Eigenvalues of hermitian operators are real. Proof : Let ( f is eigenfunction of Q with eigenvalue q ) ( Q is hermitian )   QED

Theorem 2 : Eigenfunctions belonging to distinct eigenvalues are orthogonal. Proof : ( f , g are eigenfunctions of Q with eigenvalue q and r ) Let ( Q is hermitian )  ( Theorem 1 ) r  q  QED Using the Gram-Schmidt orthogonalization scheme on the degenerate subspaces, all eigenfuctions of a hermitian operator can be made orthonormal.

Axiom (Dirac) : Eigenfunctions of an observable operator are complete. ( Required to guarantee every measurement has a result. ) Note: Eigenfunctions of a hermitian operator on a finite dimensional space are complete. Not necessarily so if the space is infinite dimensional.

3.3.2. Continuous Spectra Example 3.2. Momentum Operator Eigenfunctions not normalizable. Example 3.2. Momentum Operator Example 3.3. Position Operator

Example 3.2. Momentum Operator Find the eigenfunctions & eigenvalues of the momentum operator. Ans. Set  Placement of the 2 is a matter of taste. ( Dirac orthogonality )

is complete ( Fourier transform ) For any real function f :  i.e.

Example 3.3. Position Operator Find the eigenfunctions & eigenvalues of the position operator. Ans. Let gy be the eigenfunction with eigenvalue y.  Dirac orthonormality :  Completeness: For any real function f with

Preferred Derivation    

3.4. Generalized Statistical Interpretation Measurement of an observable Q(x, p) on a state (x, t) always gets one of the eigenvalues of the hermitian operator Q(x, i d / dx). 2.a) Discrete eigenvalues, orthonormalized eigenfunctions : Probability of getting the eigenvalue qn is 2.b) Continuous eigenvalues, Dirac orthonormalized eigenfunctions : Probability of getting an eigenvalue q(z) with z between z and z + dz is 3. Upon measurement,  collapses to fn or fz .

Proof for Discrete Eigenvalues  If  is normalized.  | cn |2 could be a probability.  | cn |2 = Probability of getting the eigenvalue qn .

Position Eigenfunctions Eigenfunction of position operator :  = probability of finding particle within ( y, y+dy ).

Momentum Eigenfunctions Eigenfunction of momentum operator : Momentum space wave function. Position space wave function. = probability of finding particle with momentum within ( p, p+dp ). Note :

Example 3.4. A particle of mass m is bound in the delta function well V(x) =   ( x). What is the probability that a measurement of its momentum would yield a value greater than p0 = m  / . Ans.

 Do Prob 3.11 Read Prob 3.12

3.5. The Uncertainty Principle Proof of the Generalized Uncertainty Principle The Minimum Uncertainty Wave Packet The Energy-Time Uncertainty Principle

3.5.1. Proof of the Generalized Uncertainty Principle System in state . A hermitian,  A  real.  where where Schwarz inequality

 A, B hermitian,  A ,  B  real.  f  g , A  B  where Generalized Uncertainty Principle  or

 Observables A and B are incompatible if Measuring A collapses state to an eigenstate of A, and similarly for B. If A and B are incompatible, repeated measurements of A, B, A, B, ..., will never, except by accident, get the same values. If A and B are compatible, repeated measurements of A, B, A, B, ..., will get the same values provided A and B are also compatible with H. Stationary states of a system can be specified by the eigenvalues of a maximal (complete) set of observables compatible with H. Do Prob 3.15 Read Prob 3.13

3.5.2. The Minimum Uncertainty Wave Packet E.g., ground state of a harmonic oscillator , Gaussian wave packets of a free particle. Minimum Uncertainty Setting the Schwarz inequality to equality  c = constant Uncertainty principle keeps only Im < f | g > to give Minimal uncertainty thus implies i.e.   a = real  or minimal uncertainty state

For position-momentum uncertainty :  Prob 3.16

3.5.3. The Energy-Time Uncertainty Principle Position-momentum uncertainty : 4-vectors: ( c t, x ), (E / c, p ) Special relativity suggests energy-time uncertainty : Non-relativistic theory : t is a parameter, not a dynamic variable. t  t . t = time for system to change appreciably.  Energy-time uncertainty is NOT like the other uncertainty pairs.

Let Q( x, p, t ) be some characteristic observable of the system. H hermitian   Q = Q( x, p ) Define 

Example 3.5. For a stationary state, for all observable Q.  E  0, t   Time dependence occurs only for linear combinations of stationary states. E.g., a, b, 1 , 2 real.  E  E2  E1 

Example 3.6. How long does it take for a free-particle wave packet to pass by a particular point ? Roughly, x  width of wave packet. Note :   c.f. Prob 3.19

Example 3.7. The  particle lasts about 1023 s before spontaneously disintegrate. A histogram of all measurements of its mass gives a bell-shaped curve centered at 1232 MeV/c2, with a width about 120 MeV/c2. Why does the rest energy ( mc2 ) sometimes come out larger than 1232, and sometimes lower? Is this experimental error? Ans. Measured data : while  Spread in measured mass is close to minimum uncertainty allowed.  It’s not experimental error.

Caution : does NOT mean you can violate energy conservation by borrowing energy E and paying it back within t   / (2E).

3.6. Dirac Notation The set of all eigenfunctions of any observable is complete. It can be used as a basis for the system’s Hilbert space. State of system ( vector in Hilbert space ) : t is a parameter r-representation : Basis = Eigenstates of the position operator. completeness orthogonality, Dirac normalization

p - Representation p-representation : Basis = Eigenstates of the momentum operator. completeness   orthogonality, Dirac normalization

Let | n  is an energy eigenstate. completeness orthonormality     where 

Operators: Discrete Basis Operators are linear transformations : Orthonormal basis   Or where matrix elements

Operators: Continuous Basis Operators are linear transformations : Dirac-orthonormal basis   Or where matrix elements If Q is diagonal, i.e., then

Example: x-representation Matrix elements : Since ( prove it ! ) we have

 

Example 3.8 Consider a system with only 2 independent states General normalized state : with Most general Hamiltonian : with h1, h2 real. Assume with h, g real. If the system starts out ( at t = 0 ) in state | 1 , what is its state at time t ? Ans : Time-dependent Schrodinger eq. Time-independent Schrodinger eq.

 Characteristic eq.   Eigenenergies are : Corresponding eigenvectors are given by:   Eigenenergies are : Normalized :

System starts out ( at t = 0 ) in state | 1  : neutrino oscillation: e   .

Dirac notation : ket : | f  is a vector in L2 . bra :  f | is in its dual L2*. For L2 infinite dimensional, | f  is a function,  f | a linear functional: so that For L2 finite dimensional, |   is a column vector,   | is a row vector : so that

Let |   be normalized, then is the projection operator onto the 1-D space spanned by |  . E.g. is a vector in the direction |   with magnitude  |  . Let { | en  ; n = 1, ..., N } be an orthonormal basis, i.e., then = unit operator in N-dimensional space spanned by { | en  } . E.g. magnitude of |  along | en  If { | ez  } is a Dirac orthonormalized continous basis, i.e., then = unit operator in function space spanned by { | ez  }. Read Prob 3.21, 3.24 Do Prob 3.22, 3.23