2. Time Independent Schrodinger Equation Stationary States The Infinite Square Well The Harmonic Oscillator The Free Particle The Delta Function Potential The Finite Square Well

1.1. Stationary States Schrodinger Equation : Assume (  is separable )  = const = E if V = V(r)  Time-independent Schrodinger eq.

Properties of Separable  1. Stationary States: Expectation values are time-independent. In particular :

2. Definite E : Hamiltonian :   Time-independent Schrodinger eq. becomes

3. Expands Any General Solution : Any general solution of the time dependent Schrodinger eq. can be written as where The cn s are determined from { n } is complete. Proof : 

Example 2.1 A particle starts out as What is ( x, t ) ? Find the probability, and describe its motion. Ans. if c &  are real. Read Prob 2.1, 2.2

2. The Infinite Square Well   0 for x  [ 0, a ] for x  [ 0, a ] General solution. Allowable boundary conditions ( for 2nd order differential eqs ) :  and   can both be continuous at a regular point (i.e. where V is finite). Only  can be continuous at a singular point (i.e. where V is infinite).

Boundary Conditions B.C.:  continuous at x = 0 and a , i.e.,    n = 0    0 (not physically meaningful).   k give the same independent solution.  E is quantized. Normalization:

Eigenstates with B.C. has a set of normalized eigenfunctions with eigenvalues n = 1 is the ground (lowest energy) state. All other states with n > 1 are excited states. even, 0 node. odd, 1 node. even, 2 nodes.

Properties of the Eigenstates Parity = () n  1 . [ True for any symmetric V ] Number of nodes  n  1. [ Universal ] Orthogonality: [ Universal ] if m  n. Orthonormality: 4. Completeness: [ Universal ] Any function f on the same domain and with the same boundary conditions as the n s can be written as

Proof of Orthonormality

Condition for Completeness where { n } is orthonormal. Then   if { n } is complete.

Conclusion Stationary state of energy is General solution: where so that

Example 2.2 A particle in an infinite well has initial wave function Find ( x, t ). Ans. Using we have

Normalization:  for n odd B = Bernoulli numbers 

In general :  { n } orthonormal  normalized If then | cn |2 = probability of particle in state n.   H   average of En weighted by the occupation probability.

Example 2.3. In Example 2.2, ( x, t ) closely resembles 1 (x) , which suggests | c1 |2 should dominate. Indeed,    where  Do Prob 2.8, 2.9

3. The Harmonic Oscillator Classical Mechanics: Hooke’s law: Newton’s 2nd law. Solution: Potential energy: parabolic Any potential is parabolic near a local minimum. where

Quantum Mechanics: Schrodinger eq.: Methods for solution: Power series ( analytic )  Hermite functions. Number space (algebraic)  a , a+ operators.

3.1. Algebraic Method Schrodinger eq. in operator form: Commutator of operators A and B : Eg.  f  canonical commutation relation Let ,  real, positive constants  

Set    Let  i.e., H , a+a and a a+ all share the same eigenstates. Their eigenvalues are related by where

Adjoint or Hermitian Conjugate of an Operator Given an operator A , its adjoint (Hermitian conjugate) A+ is defined by  ,   Proof : ,  real & positive Integration by part gives: ( ,   0 at boundaries )   Note:

n-Representation Consider an operator a with the property Define the operator with its eigenstates and eigenvalues Meaning of a+ n :  a+ n is an eigenstate with eigenvalue larger than that of  n by 1. a+  raising op Meaning of a n :  a n is an eigenstate with eigenvalue smaller than that of  n by 1. a  lowering op Let N be bounded below ( there is a ground state 0 with eigenvalue 0 ).   i.e. Hence with

Normalization Let ( ,   normalization const. ) with  n  Assuming ,  real :

Orthonormality  for m  n

Harmonic Oscillator : n-Representation  a ~ a , a+ ~ a+ , a+ a ~ a+ a   Equation for 0 : 

0  Set A real :  Normalized

Example 2.4 Find the first excited state of the harmonic oscillator. Ans: 

Example 2.5 Find  V  for the nth state of the harmonic oscillator. Ans:   Do Prob 2.13 (drudgery)

3.2. Analytic Method is solved analytically. Set  Set where 

Asymptotic Form For x,     as    not normalizable Set h solved by Frobenius method (power series expansion)

h()  Asume    recursion formula

Termination Even & odd series starting with a0 & a1 , resp. For large j :   explodes as    Power series must terminate. Set   j  n+2 

 with E  En

Hermite Polynomials Hermite polynomials : n even : set a0  1 , a1  0 n odd : set a0  0 , a1  1 an  2n Normalized:

n n  3 n  2 n  1 n  0

| 100 |2 ,  (x) Classical distribution : A = amplitude Do Prob 2.15, 2.16 Read Prob 2.17

4. The Free Particle  Eigenstate: Stationary state: to right to left phase velocity k > 0 : to right k < 0 : to left Reset:

Oddities about k (x,t)  Classical mechanics :  not normalizable k is not physical (cannot be truly realized physically ). k is a mathematical solution ( can be used to expand a physical state or as an idealization ).

Wave Packets General free particle state : wave packet Fourier transform

Example 2.6 A free particle, initially localized within [ a, a ], is released at t  0 : A, a real & positive Find ( x, t ). Ans: Normalize ( x, 0 ) :  

FT of Gaussian is a Gaussian.

Group Velocity In general : (k) = dispersion For a well defined wave packet,  is narrowly peaked at some k = k0 . To calculate ( x, t ), one need only  (  moves with velocity 0 . ) Define group velocity 3-D Free particle wave packet :  Reminder: phase velocity Do Prob 2.19 Read Prob 2.20

Example 2.6 A Consider a free particle wave packet with  (k) given by A, a real & positive Find ( x, t ). Ans: Normalize ( x, 0 ) :  

t = { 0, 1, 2 }ma2/

5. The Delta Function Potential Bound States & Scattering States The Delta Function Well

5.1. Bound States & Scattering States Classical turning points x0 : If V(r)  E for r  D and V(r) > E for r  D, then the system is in a bound state for r  D. If V(r)  E everywhere, then the system is in a scattering state. Quantum systems (w / tunneling) : If V() > E, then the system is in a bound state. If V() < E, then the system is in a scattering state. If V() = 0, then E < 0  bound state. E > 0  scattering state.

The Delta Function Dirac delta function : such that  f , a, and c > b  is a generalized function ( a distribution). [ To be used ONLY inside integrals. ] Setting f (x) = 1 gives Rule : ( meaningful only inside integrals.) Proof :

2. The Delta Function Well  For x  0 : Bound states : E < 0 Scattering states : E > 0 Boundary conditions : 1.  continuous everywhere. 2.   continuous wherever V is finite.

Bound States For x  0 : Bound states ( E < 0 ) : Set   real & positive (  ) = 0   continuous at x  0  B  C

Discontinuity in   (x0) is discontinuous at V(x0)  . Let   continuous & finite   

   Only one bound state Normalization :  

Scattering States Scattering states ( E > 0 ) : Set  for x  0 k real & positive  continuous at x  0  Let 

2 eqs., 4 unknown, no normalization. Scattering from left : A : incident wave C : transmitted wave B : reflected wave D = 0 x  Reflection coefficient Transmission coefficient

Delta Function Barrier  No bound states For the scattering states, results can be obtained from those of the delta function well by setting   .  Note: Since E < V inside the barrier, T  0 is called quantum tunneling. ( T = 0 in CM) Also: In QM, R  0 even if E > Vmax . ( R = 0 in CM) Do Prob 2.24 Read Prob 2.26

6. The Finite Square Well

Bound States ( E < 0 )  l &  are real & positive  finite  Symmetry :   If (x) is a solution, (x) is also a solution.

Even Solutions  continuous at a :  or   continuous at a :    where  or

Graphic Solutions E < 0  z < z0

Limiting Cases 1. Wide, deep well ( z0 large ) : Lower solutions :  c.f. infinite well 2. Shallow, narrow well ( z0 <  / 2 ) : Always 1 bound state.

Scattering States ( E > 0 )  l & k are real & positive

Scattering from the Left  continuous at a :   continuous at  a :  continuous at a :   continuous at a : Eliminate C, D and express B , F in terms of A ( Prob 2.32 ) :

T  1 when ( well transparent ) i.e. ( at energies of infinite well ) ( Ramsauer-Townsend Effect ) Do Prob 2.34