130_notes.dvi

Use the commutator relation betweenAandA†to derive [H,A]. Now show thatAis the lowering operator for the harmonic oscillatorene ...

11 More Fun with Operators 11.1 Operators in a Vector Space 11.1.1 Review of Operators First, a little review. Recall that thesq ...

Note that we can simply describe thejtheigenstate at|j〉. Expanding the vectors|φ〉and|ψ〉, |φ〉 = ∑ i bi|i〉 |ψ〉 = ∑ i ci|i〉 we can ...

This is an extremely useful identity for solving problems. We could already see this in the decom- position of|ψ〉above. |ψ〉= ∑ i ...

If we also consider the spin of the electron in the Hydrogen atom, wefind that we need to add one more commuting operator to lab ...

Plug in thatλ. (∆A)^2 − 1 4 〈ψ|[U,V]|ψ〉^2 (∆B)^2 + 〈ψ|[U,V]|ψ〉^2 2(∆B)^2 ≥ 0 (∆A)^2 (∆B)^2 ≥− 1 4 〈ψ|[U,V]|ψ〉^2 =〈ψ| i 2 [U,V]|ψ ...

which becomes simple if the operator itself does not explicitly depend on time. d dt 〈ψ|A|ψ〉= i ̄h 〈ψ|[H,A]|ψ〉 Expectation value ...

11.6 The Heisenberg Picture* To begin, lets compute the expectation value of an operatorB. 〈ψ(t)|B|ψ(t)〉 = 〈e−iHt/ ̄hψ(0)|B|e−iH ...

11.7 Examples 11.7.1 Time Development Example Start off in the state. ψ(t= 0) = 1 √ 2 (u 1 +u 2 ) In the Schr ̈odinger picture, ...

state of the particle and on the potential. Answer d〈A〉 dt = 1 i ̄h 〈[A,H]〉 d〈x〉 dt = 1 i ̄h 〈[ x, p^2 2 m ]〉 = 1 i ̄h 〈 p m ( − ...

12 Extending QM to Two Particles and Three Dimensions 12.1 Quantum Mechanics for Two Particles We can know the state of two part ...

12.2 Quantum Mechanics in Three Dimensions We have generalized Quantum Mechanics to include more than one particle. We now wish ...

∇~ 2 =−~∇r+ m^2 m 1 +m 2 ∇~R Putting this into theHamiltonianwe get H= − ̄h^2 2 m 1 [ ~∇^2 r+ ( m 1 m 1 +m 2 ) 2 ∇~^2 R+^2 m^1 m ...

12.4 Identical Particles It is not possible to tell the difference between two electrons. Theyare identical in every way. Hence, ...

So forNparticles filling the levels, (n^2 x+n^2 y+n^2 z) (^32) max= 6 N π . (n^2 x+n^2 y+n^2 z)max= ( 6 N π )^23 The energy corr ...

13 3D Problems Separable in Cartesian Coordinates We will now look at the case ofpotentials that separate in Cartesian coordinat ...

13.1.1 Filling the Box with Fermions If we fill a cold box withNfermions, they will all go into different low-energy states. In ...

We can nowrelate the Fermi energy to the number of particles in the box. EF= π^2 ̄h^2 2 mL^2 rn^2 = π^2 ̄h^2 2 mL^2 ( 3 N π )^23 ...

To understand the collapse of stars, we must compare this to thepressure of gravity. We compute this approximately, ignoring gen ...

from the electrons grows faster than the pressure of gravity, the star will stay at about earth size even when it cools. If the ...