Introduction to quantum mechanics#
I will start with a quasi-historical, phenomenological discussion of some basic aspects of quantum mechanics:
The need for a new dimensionful quantity
with units [Energy time].The quantization of the energy of light with frequency
into parcels (“photons”) with energy ,The fact that quantum states can be written as complex vectors, and that the sum of two such vectors is another legitimate quantum state.
The probabilistic nature of the outcome of quantum measurements.
These are all important; point (3) motivates a serious survey/review of linear algebra, which is also needed to discuss (4) more precisely.
Blackbody radiation#
Consider a cavity with walls at some temperature
What is the expected energy density inside the cavity? We could (following many textbooks) try to comute this from classical electromagnetism, but we will appeal to dimensional analysis.
The total energy density should be an integral over all allowed frequencies:
Here we are assuming that the volume of the box only appears in the total energy
We have the following dimensionful quantities available to construct
The frequency
.The temperature writen as a thermal energy
.The speed of light
.
To get an energy density we can make an energy from
where
The total energy is thus
This is sometimes termed the “ultraviolet catastrophe”. What is needed is some new dimensionful scale, so that the integral is better behaved. One possibility is to just assume that the electromagnetic field has a largest possible frequency
However, the real issue is that the black body spectrum does not look like this.
Planck deduced a functional form that he later justifies with a hypothesis. We well cheat and start with that hypothesis. Let us fix the polarization state and the wavenumber
whefre
where we have used the classic formula
If we carefully sum over all wavenumbers (with fixed frequency) and polarizations, we get:
One can show that this takes the Rayleigh form for
where
In other words, the fact that we can create a new energy scale
This black body spectrum is observed with exquisite prediction. As an example, a prediction of the hot big bang theory is that the early universe had a phase in which electrons, protons, and photons were in a state of thermal equilibrium with temperature
The photoelectric effect#
The next phenomenon was famously discussed by Einstein during his annus mirabilis of 1905. Consider a beam of light with frequency
Hertz observed the following in 1887:
The plates emit electrons (and no positvely charged particles)
Whether the plate emits electrons depends only on the frequency of the incoming light, and not on its intensity.
The magnitude of tte current in the cathode is proportional to the frequency
The energy of each photoelectron (as measured by mmeasuring for what
an electron hits the cathode) is independent of the intensity of the light, and is linear in frequency, above some critical frequency which depends on the metal (and not on the photon frequency).
Einstein’s interpretation was that light consistened of single particles or \photons, each of which carries an energy
where
Now consider an electromagnetic field with a given polarization and wave vector
Photon polarization#
Classical description#
Consider an electromagnetic field propagating in a vacuum along the
where
The magnetic field
This supports various polarization states such as
Linear/plane polarization:
where is a positive real number and a unit vector in the plane. In particular correspond to plane polarization along the - and -axis respectively.Circular polarization:
Left circular polarization (LCP): \(E_y = - i E_x = - i E e^{i\delta)\), or equivalently
. Thus:
Right circular polarization (RCP):
or . Thus:
Note that for any plane wave,
The energy density is
Assuming the field propagates in some volume
the factor of 2 in the denominator comes from integrating the
Now let us pass our beam through a polarizer. We call an “x-polarizer” one which admits only light polarized along the
For a polarizer aligned along
This generally reduces the energy of the beam of light. For example, we can show that if we consider light polarized along
The energy is in this case cut in half:
Quantum description#
If we apply Planck and Einstein’s insights, the beam of light consists of photons with energy
which photons pass through the polarizer?
also what if
is odd?
Our best available interpretation is that each photon has a
For more general polarizations,
So the probability of a single photon passing through the polarizer, given the polarization state listed above, is
<<<<<<< HEAD
For a more general polarization states, the probability of the photon passing through the polarizer aligned along the
For single photon with fixed frequency and wavenumber,
For an electric field satisfying this relation, we can write a “state vector” that describes the photon polarization:
The notation
Finally, not that there is a linear structure to the space of photon polarizations, in that we can add two polarization vectors and get another polarization vector, up to the overall normalization constraint (135) for a single photon. In particular, if we define
then any polarization can be described by the linear combination
with
You can show that any polarization state satisfying (135) can be written as
with
The essential point here is that the states of the photon are described by a 2-coponent vector; these vectord can be added to describe other polarization states; and the components are related by taking the absolute value squared to the probability of certai exoperiments yielding a certain outcome.
This hints at a very general structure for describing physical states and teh reults of mesurement in quantum mechanics. To set this up we need to lay down the correct mathematical language for describing such vectors, namely linear algebra.