My semester as a quantum mechanic

Hello all,

I have one class left before having finished my undergraduate coursework in quantum mechanics.  I would like to share my experience in that course and some of what I have learned in that course with you all.  I will be describing the topics we learned in some level of detail, so those of you who have no experience with quantum mechanics can know some of what to expect here, but without the mathematics that our friend Ateixeira intends on going into.

The assigned text, Gasiorowicz, was poor as a pedagogical book in my opinion.  I had to look elsewhere.  I made extensive use of the book by Nouredine Zettili which was fantastic from a pedagogical standpoint.  I also used Schaum’s Outline of Quantum Mechanics as a supplement.  I know many in my class used Griffiths to supplement the class text, which is what I believe we will be following here, and I heard many great things about it, though I did not use it personally.  Despite having a poor text, I currently have an A in the class, and if I do reasonably well on my final that should not change.

So lets get down to the actual physics.  Here is a list of things that we covered in my class:

  • The hydrogen atom
  • Mathematical formalism of quantum mechanics
  • The harmonic oscillator in the number representation
  • Orbital angular momentum
  • Spin angular momentum
  • Addition of angular momentum
  • Time independent perturbation theory
  • The application of perturbation theory to the hydrogen atom
  • The variational method and its application to the helium atom (To come on Monday, November 28)

Now, before describing any of these topics, I’d like to briefly talk about quantum mechanics in general.  My professor sort of assumed we had a working knowledge of the postulates of quantum mechanics, and the 1-dimensional Schrodinger equation and its application.  On the first day of class he presented the three dimensional Schrodinger equation and over the next three weeks proceeded to solve it for a central potential (the hydrogen atom).  I did not have a working knowledge of the Schroedinger equation, so I had a lot of catching up to do.  Thankfully, for this there was Zettili.  Because of my struggles to understand the material and the beginning of the semester, I don’t want to simply jump into what my coursework was before discussing quantum mechanics in general as an aside.

Quantum mechanics, as presented to me, is the study of the Schrodinger equation.  It comes in many forms, though the most familiar is probably the time-independent Schrodinger equation in the position representation:

\frac{\hbar^{2}}{2m} \nabla^{2} \psi \left( r \right) + V(r) \psi (r) = E \psi (r).

Quantum theory gives us a stochastic description of nature, and does away with Newtonian determinism.  From a philosophical perspective, this to me is the most stunning implication of quantum mechanics, though I do not wish to take much time in discussing philosophy here.  In quantum mechanics, a particle is completely described by its wavefunction \psi (r).  Observable information about a particle can be extracted from the wavefunction by applying an operator to it.  In the Schrodinger equation above, \frac{\hbar^{2}}{2m} \nabla^{2} is the kinetic energy operator whereas V(r) represents a generic potential energy operator.  E is a numerical value for energy.  Basically, the version of the Schrodinger equation I have presented above allows you to calculate the possible energy values that a particle is allowed to take for a given potential.  Once the differential equation is solved for the wavefunction and the energy values that its allowed to take the probability of finding the particle at any of those energy values is easily computed.  As mentioned, this is not the only version of the Schrodinger equation, and others are more appropriate for different physical situations.  I should mention, however, that there are a very limited number of cases that the Schrodinger method is exactly solvable, and for most physical situations approximation techniques need to be applied.

With this information, I think we’re ready to proceed on to the hydrogen atom.  This is one of the few physical situations for which the Schrodinger equation is exactly solvable.  The most common isotope of hydrogen consists of a proton and an electron.  The proton exerts a central Coulomb force on the electron.  Associated with this force is the Coulomb potential, which becomes V(r) in the Schrodinger equation:

V(r) = \frac{-e^{2}}{4 \pi \epsilon_0 r}.

To find solutions to the Schrodinger equation for the electron with the above potential the Schrodinger equation is represented in spherical coordinates since the Coulomb potential is isotropic, and separation of variables is applied.  The calculation is involved, and took the professor about 7 hours to solve, however there were some interesting results that emerged.  The wavefunction solutions to the hydrogen atom are known as orbital.  Three numbers, known as quantum numbers, define each orbital.  There is the principal quantum number which is associated with the energy associated with an orbital, the angular momentum quantum number which determines the magnitude of the angular momentum of an orbital, and the magnetic quantum number which determines the projection of the angular momentum operator.  Once the wavefunctions that solve the Schrodinger equation for the Coulomb potential are known, the energy spectrum of the hydrogen atom is also known.  When an electron transitions from one orbital to another, the energy that it has lost is emitted as a photon.  Since the energy associated with a photon determines the color of the photon, knowing the wavefunctions of the hydrogen atom allow us to calculate the entire spectrum of the hydrogen atom based on the electron’s transitions between orbitals.  Of everything I have learned, I believe this is the most striking demonstration of the power of quantum theory.  With it we can now explain the entire hydrogen spectrum.

I should stress here that though the Schrodinger equation is exactly solvable for the Coulomb potential, that does not mean we have a complete description of the hydrogen atom.  The Schrodinger equation is nonrelativistic.  Though the relativistic corrections to the Schrodinger equation are minor, they are important.  For instance, solutions to the Schrodinger equation with the Coulomb potential do not say anything about spin angular momentum.  To get a full picture of spin, relativity needs to be considered.  There are other corrections as well, some of which I will discuss later.

After finishing our discussion of the hydrogen atom, the professor went ahead to introduce the mathematical formalism of quantum mechanics.  This is where we learn about kets and bras; abstract representations of the wavefunction.  We also learned some about the Hilbert space, the abstract, infinite dimensional vector space upon which quantum mechanics is performed.  We learned a little bit about commutators.  We learned about different types of operators, and how they can change wavefunctions.  We learned about a specific class of operators in particular, known as Hermitian operators, which are associated with every physical observable.  We learned about the Schrodinger perspective and the Heisenberg perspective on how a quantum system evolves through time.  We also proved the Heisenberg uncertainty principle in a much more general fashion.  There may be things I am forgetting.  In this portion of the course we didn’t learn much about physics itself, however we learned a lot about the background mathematics necessary for understanding quantum mechanics.  Since I suppose describing physics is a lot easier than describing abstract mathematical ideas, I will leave this section as is for now, since this section of the course can not really be done justice without introducing the mathematics itself.

Next we looked at the quantum harmonic oscillator.  The harmonic oscillator has a potential of \frac{1}{2} m \omega^{2} \hat{x}^2, where \hat{x} is the position operator.  We approached this using what our professor termed the number basis.  We found an energy spectrum of the quantum oscillator to be

E_n = \hbar \omega \left( n + \frac{1}{2} \right).

One of the most interesting results in our analysis of the quantum oscillator is the existence of zero-point energy, the amount of energy the oscillating particle has in its ground state (when n=0).  The ground state energy of the quantum oscillator is nonzero unlike its classical counterpart.  There were two operators introduced in our analysis of the quantum harmonic oscillator: the creation operator and the annihilation operator.  The annihilation operator, when acting on a wavevector lowered its energy by 1 state, and the creation operator raised the energy of the wavevector by 1 state.  I believe we studied this in preparation for a more detailed study of angular momentum, which used ideas very similar to creation and annihilation.

In classical mechanics angular momentum is defined as \vec{L} = \vec{r} \times \vec{p}, where r is the position of the particle with respect to some origin and p is the linear momentum of the particle.  In quantum mechanics, the angular momentum is defined in the same way, except r and p are replaced with the position operator and momentum operator respectively.  Based on this we were able to conclude that

[L_i, L_j] = L_i L_j - L_j L_i = i \varepsilon_{ijk} L_k.

It turns out this equation allows integer values and half-integer values for the angular momentum.  The integer values are associated with orbital angular momentum.  As it turns out, there is a physical quantity with half-integer angular momentum: spin angular momentum for particles known as fermions, such as electrons.  This is a purely quantum quantity, without any real classical analogue.  Furthermore, spin cannot be derived from first principles without accounting for relativity.  We investigated angular momentum in some mathematical detail using tools very similar to those developed for the quantum oscillator.  We also investigated spin operators represented as what is known as Pauli matrices.

As I mentioned before, very few problems in quantum mechanics are solvable exactly.  There are two approximation methods learned in my class which allow us to say something about a physical system that is not exactly solvable.  We are currently working on the variational method.  This method allows us to put an upper bound on the lowest energy expectation value for a particle, and make estimations for some excited states.  We will be applying this to the helium atom today in class.

The other method we have learned for approximating quantum systems is known as perturbation theory.  If we have a system that varies slightly from an exactly solvable system, perturbation theory can be used to approach it.  One application of perturbation theory is in the study of the Stark effect.  Here, a hydrogen atom is placed in an external electric field.  In this case, the spectrum of the hydrogen atom shifts and splits.  Perturbation theory allows you to approach this effect from an analytical perspective.  Perturbation theory also allows us to make some relativistic corrections to the hydrogen atom, correct for the interaction between the magnetic field of the proton and the spin of the electron, and other corrections to the hydrogen atom that the Schrodinger equation could not completely account for.

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