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 ifV = 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
0for 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 fon 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 ofEn 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: Hookes law: Newtons 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 ofa+ n: a+ nis an eigenstate with eigenvalue larger than that of nby 1. a+ raising op Meaning ofa n: a nis an eigenstate with eigenvalue smaller than that of nby 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 form 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 : seta0 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
kis not physical (cannot be truly realized physically ). kis 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 velocity0 . ) Definegroup 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 pointsx0 : If V(r) Eforr D andV(r) > Eforr D, then the system is in a bound state forr D. If V(r) Eeverywhere, 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. ] Settingf (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 atx 0B 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
kreal & positive continuous atx 0 Let 2 eqs.,4 unknown,no normalization.
Scattering from left : A :incident waveC :transmitted wave B :reflected waveD = 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 < Vinside the barrier, T 0 is called quantum tunneling. ( T = 0 in CM) Also:In QM,R 0even 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 < 0z < 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, Dand express B , F in terms of A( Prob 2.32 ) : T when ( well transparent ) i.e. ( at energies of infinite well ) ( Ramsauer-Townsend Effect ) Do Prob 2.34