Modern Physics Ch.7:H atom in Wave mechanics

Methods of Math. Physics, 27 Jan 2011, E.J. Zita

• Schrödinger Eqn. in spherical coordinates

• H atom wave functions and radial probability densities

• L and probability densities

• Spin

• Energy levels, Zeeman effect

• Fine structure, Bohr magneton

Recall the energy and momentum operators

hcE pc

hp

From deBroglie wavelength, construct a differential operator for momentum:

2

h

2k

2

2

hhp k i

x

Similarly, from uncertainty principle, construct energy operator:

E t E it

Energy conservation Schrödinger eqn.

E = T + V

E = T + Vwhere is the wavefunction and

operators depend on x, t, and momentum:

p̂ ix

E i

t

2 2

22i U

t m x

Solve the Schroedinger eqn. to find the wavefunction, and you know *everything* possible about your QM system.

Schrödinger EqnWe saw that quantum mechanical systems can be described by wave functions Ψ.

A general wave equation takes the form:

Ψ(x,t) = A[cos(kx-ωt) + i sin(kx-ωt)] = e i(kx-ωt)

Substitute this into the Schrodinger equation to see if it satisfies energy conservation.

Derivation of Schrödinger Equation

i

Wave function and probability

Probability that a measurement of the system will yield a result between x1 and x2 is 2

2

1

( , )x

x

x t dx

Measurement collapses the wave function

•This does not mean that the system was at X before the measurement - it is not meaningful to say it was localized at all before the measurement.

•Immediately after the measurement, the system is still at X.

•Time-dependent Schrödinger eqn describes evolution of after a measurement.

Exercises in probability: qualitative

Uncertainty and expectation values

Standard deviation can be found from the deviation from the average:

But the average deviation vanishes:

So calculate the average of the square of the deviation:

Last quarter we saw that we can calculate more easily by:

j j j

0j

22 j

22 2j j

Expectation values

2( , )x x x t dx

Most likely outcome of a measurement of position, for a system (or particle) in state :

Most likely outcome of a measurement of position, for a system (or particle) in state :

*d xp m i dx

dt x

Uncertainty principle

Position is well-defined for a pulse with ill-defined wavelength. Spread in position measurements = x

Momentum is well-defined for a wave with precise. By its nature, a wave is not localized in space. Spread in momentum measurements = p

We saw last quarter that x p

Applications of Quantum mechanics

Blackbody radiation: resolve ultraviolet catastrophe, measure star temperatures

Photoelectric effect: particle detectors and signal amplifiers

Bohr atom: predict and understand H-like spectra and energies

Structure and behavior of solids, including semiconductors

Scanning tunneling microscope

Zeeman effect: measure magnetic fields of stars from light

Electron spin: Pauli exclusion principle

Lasers, NMR, nuclear and particle physics, and much more...

Stationary states

If an evolving wavefunction (x,t) = (x) f(t)

can be “separated”, then the time-dependent term satisfies

Everyone solve for f(t)=

Separable solutions are stationary states...

i

1 dfi E

f dt

2 2

2

( , ) ( , )( , ) ( , )

2

x t x ti V x t x t

t m x

Separable solutions:

(1) are stationary states, because

* probability density is independent of time [2.7]

* therefore, expectation values do not change

(2) have definite total energy, since the Hamiltonian is sharply localized: [2.13]

(3) i = eigenfunctions corresponding to each allowed energy eigenvalue Ei.

General solution to SE is [2.14]

2 2( , ) ( )x t x

2 0H

1

( , )ni E t

n nn

x t c e

Show that stationary states are separable:Guess that SE has separable solutions (x,t) = (x) f(t)

sub into SE=Schrodinger Eqn

Divide by (x) f(t) :

LHS(t) = RHS(x) = constant=E. Now solve each side:

You already found solution to LHS: f(t)=_________

RHS solution depends on the form of the potential V(x).

t

2

2x

2

22

i Vt x

2 2

22

dV E

m dx

Now solve for (x) for various U(x)

Strategy:

* draw a diagram

* write down boundary conditions (BC)

* think about what form of (x) will fit the potential

* find the wavenumbers kn=2

* find the allowed energies En

* sub k into (x) and normalize to find the amplitude A

* Now you know everything about a QM system in this potential, and you can calculate for any expectation value

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

What is probability of finding particle outside?

Inside: SE becomes

* Solve this simple diffeq, using E=p2/2m,

* (x) =A sin kx + B cos kx: apply BC to find A and B

* Draw wavefunctions, find wavenumbers: kn L= n

* find the allowed energies:

* sub k into (x) and normalize:

* Finally, the wavefunction is

2 2

22

dE

m dx

22

2

( ) 2,

2n

nE A

mL L

2( ) sinn

nx x

L L

Square well: homework

Repeat the process above, but center the infinite square well of width L about the point x=0.

Preview: discuss similarities and differences

Infinite square well application: Ex.6-2 Electron in a wire (p.256)

Summary:

• Time-independent Schrodinger equation has stationary states (x)

• k, (x), and E depend on V(x) (shape & BC)

• wavefunctions oscillate as eit

• wavefunctions can spill out of potential wells and tunnel through barriers

That was mostly review from last quarter.

Moving on to the H atom in terms of Schrödinger’s wave equation…

Review energy and momentum operators

Apply to the Schrödinger eqn:

E(x,t) = T (x,t) + V (x,t)

p̂ ix

E i

t

2 2

22i V

t m x

1

( , )ni E t

n nn

x t c e

Find the wavefunction

for a given potential V(x)

Expectation values

2 2 *( , )x x x t dx where

Most likely outcome of a measurement of position, for a system (or particle) in state x,t:

Order matters for operators like momentum – differentiate (x,t):

*d xp m i dx

dt x

*f f dx

H-atom: quantization of energy for V= - kZe2/r

Solve the radial part of the spherical Schrödinger equation (next quarter):

Do these energy values look familiar?

Continuing Modern Physics Ch.7:H atom in Wave mechanics

Methods of Math. Physics, 10 Feb. 2011, E.J. Zita

• Schrödinger Eqn. in spherical coordinates

• H atom wave functions and radial probability densities

• L and probability densities

• Spin

• Energy levels, Zeeman effect

• Fine structure, Bohr magneton

Spherical harmonics solve spherical Schrödinger equation for any V(r)

You showed that 210 and 200 satisfy Schrödinger’s equation.

H-atom: wavefunctions (r,) for V= - kZe2/r

R(r) ~ Laguerre Polynomials, and the angular parts of the wavefunctions for any radial potential in the spherical Schrödinger equation are

( , , ) ( ) ( , )

( , )lm

lm

r R r Y where

Y spherical harmonics

Radial probability density22

,( ) ( )nP r r R r

Look at Fig.7.4. Predict the probability (without calculating) that the electron in the (n,l) = (2,0) state is found inside the Bohr radius.

Then calculate it – Ex. 7.3. HW: 11-14 (p.233)

H-atom wavefunctions ↔ electron probability distributions:

l = angular momentum wavenumber

Discussion: compare Bohr model to Schrödinger model for H atom.

ml denotes possible orientations of L and Lz (l=2)

Wave-mechanics L ≠ Bohr’s n

HW: Draw this for l=1, l=3

QM H-atom energy levels: degeneracy for states with different and same energy

Selections rules for allowed transitions: n = anything (changes in energy level)

l must change by one, since energy hops are mediated by a photon of spin-one:

l = ±1

m = ±1 or 0 (orientation)DO #21, HW #23

Stern-Gerlach showed line splitting, even when l=0. Why?

l = 1, m=0,±1 ✓ l = 0, m=0 !?

Normal Zeeman effect “Anomalous”

A fourth quantum number: intrinsic spin

L ( 1), S ( 1)Since l l let s s

If there are 2s+1 possible values of ms,

and only 2 orientations of ms = z-component of s (Pauli),

What values can s and ms have?

HW #24

Wavelengths due to energy shifts

_________

________

hcE

dE d

E

Spinning particles shift energies in B fields

Cyclotron frequency: An electron moving with speed v perpendicular to an external magnetic field feels a Lorentz force: F=ma

(solve for =v/r)

Solve for Bohr magneton…

Magnetic moments shift energies in B fields

Spin S and orbit L couple to total angular momentum J = L + S

Spin-orbit coupling: spin of e- in magnetic field of pFine-structure splitting (e.g. 21-cm line)

(Interaction of nuclear spin with electron spin (in an atom) → Hyper-fine splitting)

Total J + external magnetic field → Zeeman effect

Total J + external magnetic field → Zeeman effect

Total J + external magnetic field → Zeeman effect

History of atomic models:

• Thomson discovered electron, invented plum-pudding model• Rutherford observed nuclear scattering, invented orbital atom• Bohr quantized angular momentum, improved H atom model. • Bohr model explained observed H spectra, derived En = E/n2 and phenomenological Rydberg constant • Quantum numbers n, l, ml (Zeeman effect)• Solution to Schrodinger equation shows that En = E/l(l+1)• Pauli proposed spin (ms= ±1/2), and Dirac derived it• Fine-structure splitting reveals spin quantum number