Wave mechanics in potentialsModern Ch.4, Physical Systems, 30.Jan.2003 EJZ

Particle in a Box (Jason Russell), Prob.12

Overview of finite potentials

Harmonic Oscillator (Don Verbeke), Prob.48

Hydrogen atom

Infinite Square well: V(0<x<L) = 0, V= outside

22

2

( ) 2,

2n

nE A

mL L

2( ) sinn

nx x

L L

Overview of finite potentialsFinite well: can spill out Tunneling through finite barriers

Harmonic oscillator: V(x) =1/2 kx2

12nE n

21

42

1(

2 !n nn

mH e

n

0 1

22

1, 2

4 2

mx

H H

H

Hydrogen atom : Bohr model

We found rn = n2 r1, En = E1/n2, where the “principle quantum number” n labels the allowed energy levels.

Discrete orbits match observed energy spectrum

Hydrogen atom: Orbits are not discrete

(notice different r scales)

Hydrogen atom: Schrödinger solutions depend on new angular momentum quantum numbers

Quantization of angular momentum direction for l=2

Magnetic field splits l level in (2l+1) values of ml = 0, ±1, ± 2, … ± l

1

12

( 1) 0,1,2,..., 1

cosz l

l l where l n

L m

EE where E Bohr ground state

n l

L

L

Hydrogen atom examples from Giancoli

Hydrogen atom examples from Giancoli

Hydrogen atom examples from Giancoli

Hydrogen atom examples from Giancoli

Summary: • You can calculate permitted states and energies from

boundary conditions• Finite wells and barriers need reflection/transmission

analysis

• Infinite square well has En~n2 E1

• Harmonic oscillator has evenly spaced E• Hydrogen atom: 3D spherical solution to Schrödinger

equation yields 3 new quantum numbers:

l = orbital quantum number

ml = magnetic quantum number

ms = spin = ±1/2