Quantum Mechanics in 3 Dimensionsphysics.wisc.edu/undergrads/courses/fall09/205/... · October 09...

October 09 Modern Physics

Quantum Mechanics in 3 Dimensions

Chapter 8


Separation of time

Look for solutions harmonic in time. Suppose the wavefunction has the form


Free particle solutions

Look for solutions that factorize into a product of functions of the independent variables:

Each term depends on a different variable so each must be a constant.


Free particle solutions


Free particle solutions For any (wave)vector k there is a solution

Any superposition of solutions is a solution so we can find standing wave solutions of fixed energy. Wavepackets, spherical waves… are constructed by superposing solutions with different |k|, E


Particle in a box Consider a particle confined to a 3 dimensional infinitely deep potential well - a “box”. Outside the wave function vanishes. Inside a harmonic solution is a product of standing waves, each a linear combination of traveling waves.


Particle in a box


“Square box” or cube

If the sides have equal length

This special case has symmetry reflected in degeneracy of energy levels.


Ground state

The probability density in a slice of space of constant z:


Excited states

For a cube, the the 121 state is a rotated 211 state.


Degeneracy of states in a cube

For equal length sides, equal quantized momentum and energy can be invested in the three physically equivalent directions.


Energy Levels The energy level spectrum is unlike the 1-d case for which E~ n2 as different directions have energy, and for equal length sides each level corresponds to several wave states. For unequal sides, the degeneracy is broken and levels separate.


Waves along a tube Example: A stretched box has large L in one direction so realtively small k and a nearly continuous energy spectrum is associated with that direction.


Leaky boxes

A box with penetrable walls will have wave solutions which leak out in three dimensions.


Applications A generic ‘gas’ is a collection of non-interacting particles. The properties derive from the spectrum of wave states just described.

Examples:

Atomic or molecular gas in a macroscopic box

Conduction electrons in a metal

Nucleons inside a nucleus

In the last two examples, the particles are individually strongly interacting but collectively the effective potential is flat.


Lessons A box is a model for a bound wave in 3-d, a crude model for an electron bound to a proton.

The solutions have features in common with atomic electron bound states

- a spectrum increasing in density with energy,

- -degeneracy reflecting symmetry,

- complex pattern of nodes and lobes of probability density


Complex number review


Spherical symmetry

We use Cartesian coordinates x,y,z to describe matter waves subject to rectangular boundary conditions.

Particle waves confined by a spherical shell (eg, nucleons in a nucleus) or spherically symmetric central force are most simply described with spherical coordinates.


Spherical coordinates


Laplace operator in spherical coordinates

Change variables:

It is a chore to prove this!

Cartesian coordinates


Angular momentum operator

The angular dependence resides in the quantity L2 which is (perhaps not surprisingly) related to orbital angular momentum.

Isolate the angular dependence of the operator


Separation of spherical variables

Write the wavefunction in terms of spherical coordinates and suppose it factorizes into a product of functions:

Insert into the time independent Schrodinger equation and isolate terms as we have before.

The separation is a bit trickier but the same idea.


Separation of spherical variables

Move r dependent terms to the left side.

The terms on the left and right depend on independent variables so both must be constant. Call it -l(l+1).


Separated equations Radial equation

Angular equation


Separation of angular equation Separate polar and azimuthal angles

The 2nd term on the left that depends on azimuth must be constant. Call the constant -m2.

Next: Solve in reverse order for


Azimuthal equation The equation for the azimuthal angle is familiar:

This azimuthal factor must be continuous as the azimuthal angle runs from 0 to pi so m must be an integer.


Polar angle equation

This (Legendre) equation has solutions for a given integer m provided l is an integer. The solutions are polynomials in cos (theta) provided |m|< l+1.


Spherical harmonics The entire angular wavefunction is conventionally expressed as a normalized “spherical harmonic” function:


Completeness

Fourier analysis: Any function f(x) over x=[0,1] can be expressed as a superposition of trig functions

Similarly, any function of spherical angles can be represented as a superposition of spherical harmonics


Angular momentum operator A plane wave has a unique momentum. Mathematically this is represented by the fact that application of the momentum operator returns a unique momentum value:

Similarly, a wave proportional to a spherical harmonic has a unique orbital angular momentum magnitude and z component:


Angular momentum values Angular momentum is quantized:


Hydrogen Atom Recall the separation of variables

Assuming the angular dependence


Radial wave equation


Effective potential

The first term is positive, an infinitely repulsive force for nonvanishing l. Postive angular momentum implies the particle can not be at the origin!

The second term is attractive and dominates at large r.

The net effective potential is a well in the radial direction


Effective potential

r

Put R = g/r


Hydrogen atom radial wavefunctions

The radial equation has a sequence of solutions Rnl labeled in order of energy by “principle quantum number n=1,2,… and angular momentum l with l<n.


Hydrogen atom complete wavefunctions and energies

The complete time independent Schrodinger equation and solution which are a radial factor Rnl multiplies by an angular factor Ylm where n=1,2,… l = 1,2,…(n-1) m = -l,…,+l

The energy of these states is independent of l and m and given by Rydberg/Bohr formula!


Spectroscopic notation


Energy level diagram The wave theory tells us that the energy levels are (other than the ground state) are degenerate.

Optical transitions connect states with


Probability density

For spherically symmetric states, the probability of an electron within a shell of radius r and thickness dr is

The probability an electron is in a volume dV is


1s Radial Probability Distribution

The curve P1s(r) representing the probability of finding the electron as a function of distance from the nucleus in a 1s hydrogen-like state. Note that the probability takes its maximum value when r equals a0/Z.


1s Cloud picture

The spherical electron “cloud” for a hydrogen-like 1s state. The shading at every point is proportional to the probability density 1s(r) 2.


S state probability distributions

For given n>0, nodes in the radial wavefunction separate concentric “shells” of probability.


P state probability distributions


Higher excitations


Axial symmetry

(a) The probability density 2112 for a hydrogen-like 2p state. Note the axial symmetry about the z-axis. (b) and (c) The probability densities (r) 2 for several other hydrogen-like states. The electron “cloud” is axially symmetric about the z-axis for all the hydrogen-like states . nlml(r)


Cartesian p states

The 2p m=0 state is directional. Linear combinations of m=+1 and m=-1 are similar, and associated with bonding.


Exotic atoms The hydrogen atom wave states apply to similar matter atoms:


Antihydrogen Antihydrogen is detected through its destruction in collisions with matter particles. Pions (four light colored dashed lines) point to the annihilation point. Similarly, the annihilation of the positron produces a distinctive back-to-back two-photon signature (two dashed tracks at 180 to one another). (Adapted from Nature, 419, 456–459, October 3, 2002.)

•  The Dutch physicist Pieter Zeeman showed the spectral lines emitted by atoms in a magnetic field split into multiple energy levels. It is called the Zeeman effect.

Anomalous Zeeman effect: •  A spectral line is split into three lines. •  Consider the atom to behave like a small magnet. •  Think of an electron as an orbiting circular current loop of I =

dq / dt around the nucleus. •  The current loop has a magnetic moment µ = IA and the

period T = 2πr / v. •  where L = mvr is the magnitude of the orbital

angular momentum.

7.4: Magnetic Effects on Atomic Spectra—The Normal Zeeman Effect

•  The angular momentum is aligned with the magnetic moment, and the torque between and causes a precession of .

Where µB = eħ / 2m is called a Bohr magneton. •  cannot align exactly in the z direction and

has only certain allowed quantized orientations.

  Since there is no magnetic field to align them, point in random directions. The dipole has a potential energy

The Normal Zeeman Effect

The Normal Zeeman Effect •  The potential energy is quantized due to the magnetic quantum

number mℓ.

•  When a magnetic field is applied, the 2p level of atomic hydrogen is split into three different energy states with energy difference of ΔE = µBB Δmℓ.

mℓ Energy 1 E0 + µBB

0 E0

−1 E0 − µBB

The Normal Zeeman Effect •  A transition from 2p to 1s.

•  An atomic beam of particles in the ℓ = 1 state pass through a magnetic field along the z direction.

• 

• 

•  The mℓ = +1 state will be deflected down, the mℓ = −1 state up, and the mℓ = 0 state will be undeflected.

•  If the space quantization were due to the magnetic quantum number mℓ, mℓ states is always odd (2ℓ + 1) and should have produced an odd number of lines.

The Normal Zeeman Effect

7.5: Intrinsic Spin •  Samuel Goudsmit and George Uhlenbeck in Holland proposed that

the electron must have an intrinsic angular momentum and therefore a magnetic moment.

•  Paul Ehrenfest showed that the surface of the spinning electron should be moving faster than the speed of light!

•  In order to explain experimental data, Goudsmit and Uhlenbeck proposed that the electron must have an intrinsic spin quantum number s = ½.

Intrinsic Spin •  The spinning electron reacts similarly to the orbiting electron in a

magnetic field. •  We should try to find L, Lz, ℓ, and mℓ. •  The magnetic spin quantum number ms has only two values,

ms = ±½. The electron’s spin will be either “up” or “down” and can never be spinning with its magnetic moment µs exactly along the z axis.

The intrinsic spin angular momentum

vector .

Intrinsic Spin •  The magnetic moment is . •  The coefficient of is −2µB as with is a consequence of theory

of relativity.

•  The gyromagnetic ratio (ℓ or s). •  gℓ = 1 and gs = 2, then

•  The z component of . •  In ℓ = 0 state

•  Apply mℓ and the potential energy becomes

no splitting due to .

there is space quantization due to the intrinsic spin.

and

7.6: Energy Levels and Electron Probabilities •  For hydrogen, the energy level depends on the principle

quantum number n.

  In ground state an atom cannot emit radiation. It can absorb electromagnetic radiation, or gain energy through inelastic bombardment by particles.

Selection Rules •  We can use the wave functions to calculate transition

probabilities for the electron to change from one state to another.

Allowed transitions: •  Electrons absorbing or emitting photons to change states when Δℓ = ±1.

Forbidden transitions: •  Other transitions possible but occur with much smaller

probabilities when Δℓ ≠ ±1.

Quantum Mechanics in 3 Dimensionsphysics.wisc.edu/undergrads/courses/fall09/205/... · October 09...

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