Modern Physics Ch.7: H atom in Wave mechanics Methods of Math. Physics, 27 Jan 2011, E.J. Zita...
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Transcript of Modern Physics Ch.7: H atom in Wave mechanics Methods of Math. Physics, 27 Jan 2011, E.J. Zita...
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