Week 2 FAQ - iSites

PS10 Week 2 FAQ
Q: What are matter waves? What is a wave? What is an eigenfunction? What is Psi?
A: At the microscopic level, matter does not exist as point-like or well-bounded objects. Matter
is diffuse, and the distribution of matter often looks like it has wavelike properties. We describe
the distribution of matter by its wavefunction, represented by Ψ(x,y,z,t). At each point in space
and time, Ψ is a complex number. If you try to measure the location of a particle, the probability
of finding it at a particular region of space with volume dx dy dz, at time t, is
| Ψ(x,y,z,t)| 2 dx dy dz. Because | Ψ(x,y,z,t)| 2 is a square of a number, it is always positive, as
required for a probability.
The fact that Ψ is a complex number—in particular that it can be either positive or
negative—means that the wavefunction can undergo constructive or destructive interference with
itself. You can take two solutions for Ψ, and they might add, and enhance the probability of
finding a particle at a spot, or they might cancel, and decrease the probability of finding a
particle at that spot.
Eigenfunctions are embodiments of the function Ψ(x,y,z,t) that have some particularly
useful properties. These properties are:
1) Eigenfunctions are states with constant energy. If Ψ for a particle is an eigenfunction
of the system, then you’ll get the same energy for the particle regardless of when you
measure that energy.
2) Eigenfunctions don’t change shape with time. That is, you can write Ψ(x,y,z,t) as a
product of two functions ψ(x,y,z) x F(t). Once you’ve specified the spatial part of an
eigenfunction, ψ(x,y,z), you know its shape for all time. This is analogous to a
simple vibration of a violin string or a drumhead. These vibrations don’t change
shape with time; only their amplitude grows and shrinks periodically. For
eigenfunctions of the Schrodinger equation, the time part is always F(t) = e -i ω t , where
h ω = E n , and E n is the energy of the eigenstate.
3) For any geometry, there will be a series of eigenfunctions (like the harmonics on a
violin string). Eigenfunctions with more zero-crossings (i.e. more wiggles), tend to
be higher in energy.
4) Any wavefunction Ψ(x,y,z,t) can be represented as a sum of eigenfunctions. This is
analogous to music or speech, where any pattern of sounds can be broken down into a
sum of pure tones.
The shape and energies of the eigenfunctions depend on the environment of the particle. A
particle in a square box has one set of eigenfunctions and energies; a particle in a round box has
a different set; a particle bound by an electrostatic attraction (e.g. an atom) has another set of
eigenfunctions; a particle bound by a spring has a different set. This is just like musical
instruments: a double bass plays one set of notes, a violin plays a different set of notes.
Much of the work in quantum mechanics involves trying to identify the eigenfunctions in
a particular geometry. We won’t do this quantitatively in class, but we’ll discuss qualitatively
the eigenfunctions you find in atoms and molecules.
Q: I understand that light must have sufficient energy to excite an electron in a molecule.
But if a photon has more energy than needed, why won’t that photon be absorbed?

A: Light only gets absorbed when its energy matches the energy difference between an occupied
and an unoccupied level. In principle, if the photon had extra energy, the excess could go into
producing a low energy photon, but this is a very improbably process. Here is an intuitive
explanation. If you think of the radiation as an oscillating electromagnetic field, absorption
occurs when the frequency of oscillation of the field matches an intrinsic resonance frequency of
electrons in the atom or molecule. The electrons are set oscillating by the field, and this sucks
energy out of the field. If the oscillations of the field are either too high frequency or too low
frequency, they will not resonate with the electrons. It's like pushing a swing: to get the swing
going you have to push it at just the right frequency.
Q: So if the energy of the light must precisely match the energy difference between an
excited and a ground state electron, why aren’t all spectral lines infinitesimally narrow?
A: We only considered the role of electronic excitation in determining an absorption spectrum.
There’s a bit more to the story. As we’ll see soon, molecules can vibrate, and the vibrational
energies are quantized too. However, the spacing of the vibrational energy levels is much closer
together than the spacing of the electronic energy levels. Each electronic state has associated
with it many vibrational states, all of slightly different energies. The molecule has so many
closely spaced energy transitions that we see them as a broadly distributed continuum.
Atoms have no vibrations, so spectral lines of atomic vapors tend to be narrow. However,
atomic lines are still broadened by the Doppler effect: in a vapor, some atoms are moving
towards the source, and some are moving away. So each atom “sees” a slightly different
frequency of the incident light. This broadens the absorption spectrum of the vapor. In gases at
very low temperatures and pressures, spectral lines can become exceedingly narrow. This occurs
in interstellar space and in some specialized physics experiments.
Q: What is the probability of finding a particle with energy intermediate between the
energies of two eigenstates (i.e. during a transition)?
A: Zero! If you measure the energy of a particle, then you will *always* get the energy of one
of the eigenstates. If the wavefunction of a particle is described by a linear combination of
eigenstates, e.g. c 1 psi 1 + c 2 psi 2 , then the probability of getting energy E 1 associated with state 1
is |c 1 | 2 and the probability of getting energy E 2 associated with state 2 is |c 2 | 2 . So if you repeat
the experiment many times, the average energy might lie between E 1 and E 2 . But each
measurement will only yield outcomes E 1 or E 2 .