Summary Of Quantum Mechanics Ideas

Summary of quantum mechanics ideas Spin In general, a state |ψi doesn’t have a projection on the z axis. But the states |z+i and |z−i do have such projections, and the amplitude for |ψi to have Lz = +¯h/2 is hz + |ψi the amplitude for |ψi to have Lz = −¯h/2 is hz − |ψi. One can specify state |ψi in the {|z+i, |z−i} basis through " # hz + |ψi |ψi“ = ” . hz − |ψi The “average” value of Lz is     ¯h ¯h + |hz − |ψi|2 − . |hz + |ψi|2 + 2 2 Position In general, a state |ψi doesn’t have a position. But √ the amplitude to be within dxA of xA is ψ(xA ) dxA . One can specify state |ψi in the position basis through |ψi“ = ”ψ(x). The “average” position is Z

+∞

−∞

|ψ(x)|2 x dx =

Z

+∞

ψ ∗ (x)xψ(x) dx.

−∞

Energy In general, a state |ψi doesn’t have an energy. But the states {|ni} (corresponding to wavefunctions {ηn (x)}) do have energies, and Z +∞ the amplitude for |ψi to have energy En is hn|ψi = ηn∗ (x)ψ(x) dx. −∞

One can specify state |ψi in the energy basis through listing all the coefficients Z +∞ Dn = hn|ψi = ηn∗ (x)ψ(x) dx. −∞

The “average” energy is Z

+∞

−∞

  ¯h2 ∂ 2 ψ(x) + V (x)ψ(x) dx. ψ ∗ (x) − 2m ∂ 2 x