130_notes.dvi

Numbering the three regions from left to right, u 1 (x) =eikx+Re−ikx u 2 (x) =Aeik ′x +Be−ik ′x u 3 (x) =Teikx Again we have ass ...

We can subtract the same equations to most easily solve forR. 2 Reika= 1 2 Teika [( k k′ − k′ k ) e−^2 ik ′a + ( k′ k − k k′ ) e ...

Atawe get C 3 e−κa=Acos(ka) +Bsin(ka) −κC 3 e−κa=−kAsin(ka) +kBcos(ka). Divide these two pairs of equations to get two expressio ...

9.7.4 Solving the HO Differential Equation* The differential equation for the 1D Harmonic Oscillator is. − ̄h^2 2 m d^2 u dx^2 + ...

Take the differential equation d^2 u dy^2 (ǫ−y^2 )u= 0 and plug u(y) =h(y)e−y (^2) / 2 into it to get d^2 dy^2 h(y)e−y (^2) / ...

The series for y^2 ey (^2) / 2 ∑y^2 n+2 2 nn! has the coefficient ofy^2 n+2equal to 2 n^1 n!and the coefficient ofy^2 nequal to ...

9.7.5 1D Model of a Molecule Derivation*. ψ(x) =    eκx x <−d A(eκx+e−κx) −d < x < d e−κx x > d κ= √ − 2 mE ̄h^2 ...

The general solution in the region (n−1)a < x < nais ψn(x) =Ansin(k[x−na]) +Bncos(k[x−na]) k=sqrt 2 mE ̄h^2 Now lets look ...

We now have two pairs of equations for then+ 1 coefficients in terms of thencoefficients. An+1 = 2 maV 0 ̄h^2 k Bncos(ka)−Bnsin( ...

9.9 Sample Test Problems A beam of 100 eV (kinetic energy) electrons is incident upon apotential stepof height V 0 = 10 eV. Cal ...

b) Determine the coefficients needed to satisfy the boundary conditions. c) Calculate the probability for a particle in the beam ...

10 Harmonic Oscillator Solution using Operators Operator methods are very useful both for solving the Harmonic Oscillator proble ...

Because the lowering must stop at a ground state with positive energy, we can show that the allowed energies are En= ( n+ 1 2 ) ...

10.2 Commutators ofA,A†andH. We will use the commutator betweenAandA†to solve the HO problem. The operators are defined to be A ...

Now, apply [H,A†] to the energy eigenfunctionun. [H,A†]un= ̄hωA†un HA†un−A†Hun= ̄hωA†un H(A†un)−En(A†un) = ̄hω(A†un) H(A†un) = ( ...

We cancompute the coefficientusing our operators. |C|^2 = 〈A†un|A†un〉=〈AA†un|un〉 = 〈(A†A+ [A,A†])un|un〉= (n+ 1)〈un|un〉=n+ 1 The ...

can be used tofind the ground state wavefunction. WriteAin terms ofxandpand try it. (√ mω 2 ̄h x+i p √ 2 m ̄hω ) u 0 = 0 ( mωx+ ...

10.6.3 The expectation value ofxin the state√^12 (u 0 +u 1 ). 1 2 〈u 0 +u 1 |x|u 0 +u 1 〉 = 1 2 √ ̄h 2 mω 〈u 0 +u 1 |A+A†|u 0 +u ...

10.6.5 The expectation value of p 2 2 min eigenstate The expectation value of p 2 2 mis 〈un| p^2 2 m |un〉 = − 1 2 m m ̄hω 2 〈un| ...

〈ψ|p|ψ〉 = −i √ m ̄hω 2 〈√ 2 3 u 0 −i √ 1 3 u 1 |A−A†| √ 2 3 u 0 −i √ 1 3 u 1 〉 = −i √ m ̄hω 2 〈√ 2 3 u 0 −i √ 1 3 u 1 |− √ 2 3 u ...