130_notes.dvi

The time independent Schr ̈odinger equation is an example of an eigenvalue equation (See section 8.1). Hψi(~x) =Eiψi(~x) If we o ...

1.10 A Particle in a Box As a concrete illustration of these ideas, we study the particle in a box (See section 8.5) (in one dim ...

or d^2 u(x) dx^2 + 2 m ̄h^2 (E−V)u(x) = 0 and thegeneral solution, forE > V, can be written as either u(x) =Aeikx+Be−ikx or u ...

1.12 The Harmonic Oscillator in One Dimension Next we solve for the energy eigenstates of the harmonic oscillator(See section 9. ...

1.14 Harmonic Oscillator Solution with Operators We can solve the harmonic oscillator problem using operator methods(See section ...

1.15 More Fun with Operators We find the time development operator (See section 11.5) by solving the equationi ̄h∂ψ∂t=Hψ. ψ(t) = ...

1.16 Two Particles in 3 Dimensions So far we have been working with states of just one particle in one dimension. The extension ...

1.17 Identical Particles Identical particles present us with another symmetry in nature. Electrons, for example, are indis- ting ...

1.19 Angular Momentum For the common problem of central potentials (See section 14.1)V(r), we use the obviousrotational symmetry ...

1.20 Solutions to the Radial Equation for Constant Potentials Solutions to the radial equation (See section 15.1) in a constant ...

1.22 Solution of the 3D HO Problem in Spherical Coordinates As and example of another problem with spherical symmetry, we solve ...

Similarly, we may define thematrix elementof an operator in terms of a pair of those orthonormal basis states Oij≡〈ui|O|uj〉. Wit ...

1.25 Spin 1/2 and other 2 State Systems The angular momentum algebra defined by the commutation relationsbetween the operators r ...

In Quantum Mechanics, the momentum operator is replaced (See section 20) in the same way to include the effects of magnetic fiel ...

We exploit the gauge symmetry in EM to show that, infield free regions, the functionfcan be simply equal to a line integral of t ...

1.28 Addition of Angular Momentum It is often required to add angular momentum from two (or more) sources (See section 21) toget ...

The total number of states is always preserved. For example if I add twoℓ= 2 states together, I get total angular momentum state ...

A problem arises in the case ofdegenerate statesor nearly degenerate states. The energy denom- inator in the last equation above ...

a state of definitej. We have assumed that the effect of the field is small compared to the fine structure corrections. We can w ...

For the hydrogen ground state we are just adding two spin^12 particles so the possible values are f= 0,1. The transition between ...