130_notes.dvi

Theeigenfunctions are orthogonal. What if two of the eigenfunctions have thesame eigenvalue? Then, our proof doesn’t work. Assum ...

As before, finite terms in the right hand integral go to zero asǫ→0, but now the delta function gives a fixed contribution to th ...

know that ∫ sin(Ax) sin(Bx)dx=sin((2(AA−−BB))x)−sin((2(AA++BB))x) Answer P = |〈u( 0 L)|u(3 0 L)〉|^2 〈u( 0 L)|u(3 0 L)〉 = ∫L 0 √ ...

8.*A particle of massmis in a 1 dimensional box of length L. The particle is in the ground state. A measurement is made of the p ...

9 One Dimensional Potentials 9.1 Piecewise Constant Potentials in 1D Several standard problems can be understood conceptually us ...

For this problem, both regions haveE > V, so we will use the complex exponential solutions in both regions. This is essential ...

Continuity of the derivative of the wavefunctionatx= 0 gives [ikeikx−ikRe−ikx]x=0= [ik′Teik ′x ]x=0 Evaluate and plug inTfrom th ...

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 ...

k′= √ 2 m(E−V 0 ) ̄h^2 E=−V 0 + n^2 π^2 ̄h^2 8 ma^2 There are analogs of this in 3D. The scattering cross section oftengoes to z ...

0 0 -V 0 Energy V(x) x E −κ C e Ae + Be κx ikx -ikx x 1 C e 3 The calculation (See section 9.7.3) shows that eitherAorBmust be z ...

Potential Well Bound States -5 -4 -3 -2 -1 0 1 2 3 4 5 -20 -15 -10 -5 0 E (hbar**2/2ma**2) f(E) sqrt tan "-cot" Try this 1D Pote ...

Inside the barrier, κ= √ 2 m(V 0 −E) ̄h^2 . x incident wave reflected E V 0 V(x) transmitted x 0 0 e + Re ikx -ikx Teikx Ae +Be ...

9.2 The 1D Harmonic Oscillator Theharmonic oscillator is an extremely important physics problem. Many potentials look like a har ...

u 0 (x) = (mω π ̄h )^14 e−mωx (^2) /2 ̄h This is a Gaussian (minimum uncertainty) distribution. Since the HO potential has a par ...

9.3 The Delta Function Potential*. Take a simple,attractive delta function potentialand look for the bound states. V(x) =−aV 0 δ ...

Putting in the formula forκin terms of the energy. − 2 mE ̄h^2 = m^2 a^2 V 02 ̄h^4 E=− ma^2 V 02 2 ̄h^2 There is only one energy ...

1D Molecule -4 -2 0 2 4 x u(x) V(x) u0(x) u1(x) Two Hydrogen atoms bind together to form a molecule with a separation of 0.74 An ...

Our states will have positive energy. This potential has a new symmetry, that atranslation by the lattice spacingaleaves the pro ...

This energy band phenomenon is found in solids. Solids with partially filledbands are conductors. Solids with filled bands are i ...

n= 0, 1 , 2 , 3 ... Theangular momentum must be quantized in units of ̄h. This will prove to be true for 3 dimensions too, howev ...