Physical chemistry course outline on intrinsic spin angular momentum.
This July I will be giving lectures on spin theory under the sponsorship of NPTEL (National Programme on Technology Enhanced Learning, India) .
The lectures will be recorded at the Indian Institute of Technology (IIT) Madras which is part of the NPTEL program. A major goal of NPTEL is to raise awareness and improve scientific and technological education throughout India by use of multimedia. I will be giving a series of lectures on basic spin theory for chemistry and physics undergraduate students who have a basis in quantum mechanics; know of spin and its importance; and want to go deeper.
I will not be presenting spin from a quantum field theory (Dirac equation) viewpoint: that is graduate physics level. Rather I will use the basic experimental data for spin (the Stern-Gerlach experiment) and show that from this, and the use of quantum mechanics, a lot follows. To be complete, and because mathematics is the language of science, I cannot avoid a fair number of equations. There is no other way to really understand and get insight. However the algebra of a spin 1/2 is very simple and certainly not beyond the level of undergraduates. Vectors and matrices are enough.
Spin, being purely quantum, is an excellent way to illustrate how to apply quantum mechanics and so gives us a way to visualize these microscopic properties. Along the way entanglement will naturally arise. Finally the role spin has played in the foundations of quantum mechanics will be discussed.
This is the tentative outline:
Name of course: The experimental and theoretical basis for spin angular momentum.
Reference: The basic background of the material covered is found in my eBook, Physical Chemistry by Laidler, Meiser and Sanctuary.
Prerequisite: A basic course in quantum mechanics.
Spin angular momentum is first encountered in introductory chemistry when the periodic table is described based upon the Pauli Exclusion and Aufbau Principles. Students are told that no two spins can have the same set of quantum numbers, and since spin has only two states, up and down, then only two spins can exist in any quantum state. Therefore spin plays a pivotal part in understanding microscopic properties of matter.
The objective of this course is to understand the physical origin of spin. The story starts back in 1922 when Stern and Gerlach discovered that passing silver atoms through an inhomogeneous magnetic field produced two spots on a screen. I will use a simulation of that experiment to obtain all the experimental results for a spin ½. The probability of a spin being deflected up or down obey Malus’ law.
Following this, the laws of quantum mechanics are applied and we find spin is indeed angular momentum; there are only two states; the operator that represents spin is a vector; and spin has well defined properties. These properties are derived. Relationship will be made to qubits and entanglement.
The Heisenberg Uncertainty relations are next applied whence it is found that only one axis of spin quantization can be measured at any instance. Dispersion is discussed in terms of pure states and their superposition.
After this, we step back and look at the bigger picture of the spin algebra and symmetry. It will be shown that spin obeys the Lie algrebra of su(2) and has the symmetry SU(2). It is then shown how spins of magnitude greater than ½ can be built up by a parentage scheme.
The relationship to the usual rotation group, SO(3) is also established.
Then the spin statistics theorem is presented and the bifurcation of Nature into bosons and fermions results.
Up to this point, the spin states are all pure or superpostions. The spin density matrix is introduced to take into account mixed states.
The coupling of spin angular momentum allows the generalized to spins of any magnitude. This leads to the Chebsch-Gordan series and the generalization of angular momentum.
Spin is a purely quantum property and has no classical analogue. It therefore plays a pivotal role in understanding the foundations of quantum mechanics. This is a long standing problem going back the famous Einstein-Bohr debates and the formulation of the Einstein, Podosky Rosen (EPR) paradox in 1935. EPR questioned the completeness of quantum mechanics: “God does not roll dice!” . This question has not yet been resolved, although the vast majority of physical scientist disagree and believe Nature to be fundamentally statistical.
The course will end with a discussion of the EPR paradox and some of the consequences of entanglement, non-locality and Bell’s theorem.
- Introduction: properties of angular momentum and introduction to spin
- Stern-Gerlach-experimental evidence for spin: Use of the simulation to obtain the experimental evidence for spin ½ . Malus’ Law.
- Apply quantum mechanics to the data: In order to account for the Stern-Gerlach experiment, the postulates of quantum mechanics are applied.
- Knowledge of states and operators: The properties of the Pauli spin matrices and the spin states in any direction.
- Represented as vectors and matrices: The algebra of a spin ½ is discussed along with general group properties.
- Heisenberg uncertainty and dispersion: Spin is used to show that only two states can be dispersion free and all other states are a superposition of these.
- Summary of spin ½ : The properties of spin are summarized to give the “Big Picture”.
- Spin density operator: allows for the more usual situation whence different spin states are mixed together. The formulation of spin forms a good introduction to density matrix theory.
- Spin statistics theorem: A discussion of bosons and fermions and their properties.
- Coupling of angular momentum: It is shown how to start from a spin ½ and build up spins, and angular momentum, of any magnitude. Entanglement is shown to arise naturally.
- Foundations of quantum mechanics and spin: A discussion of the role of spin in the foundations of quantum mechanics and the famous EPR (Einstein, Podolsky, Rosen) paradox.
Have I omitted anything important?