Lecture Notes: Angular Momentum and Exchange Symmetry

Addition of Angular Momentum

• When dealing with multiple sources of angular momentum, such as two electrons each possessing their own orbital angular momentum, the combination of these momenta is described by the total angular momentum operator, denoted as j. The eigenfunctions of this operator represent the eigenstates of the system.

Example

• Consider a scenario involving two electrons, where the first electron has an angular momentum l1 = 3 and the second has l2 = 1. When these electrons do not interact, their states can be fully described by four quantum numbers: l1, m{l1}, l2, and m{l2}. Alternatively, the set of quantum numbers j, mj, l1, and l_2 can be used to represent these states.

• However, when the electrons interact, the eigenstates of the system are no longer simple products of individual electron states characterized by m{l1} and m{l2}. Although the total angular momentum is conserved, the angular momentum of each individual electron is not. This is because the interaction introduces torques that can change the individual angular momenta while keeping the total constant.

• The quantum numbers j, mj, l1, and l2 are particularly useful when describing interacting electrons. In such cases, j represents the total angular momentum, mj is its z-component, and l1 and l2 remain good quantum numbers because the magnitudes of the individual angular momenta are still conserved.

• In quantum mechanics, a "good" quantum number is one that corresponds to a conserved quantity. Conservation laws are fundamental, where energy is conserved in a closed system, momentum is conserved in the absence of external forces, and angular momentum is conserved in the absence of external torques.

• When electrons interact, they experience Coulomb forces. As a result, the eigenstates of the system are best described by the quantum numbers j, mj, l1, and l_2. Although the individual electron angular momenta change due to the interaction, their total angular momentum remains constant.

• Mathematically, the conservation of individual angular momenta can be expressed by the commutation relations. The j^2 operator (representing the square of the total angular momentum) commutes with the individual angular momentum operators l1^2 and l2^2, which means that the quantum numbers associated with these operators (i.e., l1 and l2) are preserved.

J Values

• The possible values of j range from l1 + l2 down to |l1 - l2|. In the example where l1 = 3 and l2 = 1, j can take the values from j = l1 + l2 = 3 + 1 = 4 down to j = |l1 - l2| = |3 - 1| = 2 . Therefore, the allowed values for j are 2, 3, and 4.

• For each value of j, there are 2j + 1 possible states, corresponding to the different possible values of m_j. Thus, for j = 4, there are 9 states; for j = 3, there are 7 states; and for j = 2, there are 5 states.

• Now, consider a specific state with j = 3 as an example: |j=3, mj = 1, l1=3, l_2=1 \rangle. This state can be expressed as a linear combination of states where the z-components of the individual angular momenta add up to +1. For example:

  • a |l1=3, m{l1}=2, l2=1, m{l2}=-1 \rangle

  • a |l1=3, m{l1}=0, l2=1, m{l2}=1 \rangle

Cone Diagrams

• By experimentally measuring the total angular momentum and its z-component, we can lock the system into a particular state. However, measuring the z-component of l_1 in such a state yields a probability distribution because the individual angular momenta are not precisely defined due to quantum uncertainty.

• The j^2 operator does not commute with the z-component of the individual angular momentum operator, [j^2, m{l1, z}] ≠ 0. This non-commutation implies that measuring j^2 changes the state of m{l1, z}, and vice versa, illustrating the uncertainty principle at play.

• Cone diagrams offer a visual way to represent these quantum states. For instance, an l = 3 state with m_l = 2 can be depicted as a cone, illustrating the possible orientations of the angular momentum vector.

• When adding a second object with a z-component of -1, the resulting combined cone visually represents the addition of angular momenta, where the z-components add together.

• The eigenvalues of l1^2 and l2^2 commute with those of j^2. Consequently, we can define states where l1 and l2 are specified, even though these individual quantities are not conserved.

Spin Matrices

• In quantum mechanics, operators for spin are represented by matrices. These matrices act on quantum states to yield information about the spin properties of particles.

• The spin matrices for a spin-1/2 particle, such as an electron, are given by:

  • S_x = \frac{\hbar}{2} \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}

  • S_y = \frac{\hbar}{2} \begin{bmatrix} 0 & -i \ i & 0 \end{bmatrix}

  • S_z = \frac{\hbar}{2} \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}

• The eigenstates of S_z are the column vectors \begin{bmatrix} 1 \ 0 \end{bmatrix} and \begin{bmatrix} 0 \ 1 \end{bmatrix}. These represent the spin-up and spin-down states along the z-axis, respectively.

• When S_z operates on a column matrix eigenstate, it returns a constant times the matrix, which gives the eigenvalues +\frac{\hbar}{2} and -\frac{\hbar}{2}. These eigenvalues correspond to the quantized spin values along the z-axis.

Matrix Multiplication

• The rule for matrix multiplication is: \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} e \ f \end{bmatrix} = \begin{bmatrix} ae + bf \ ce + df \end{bmatrix}

• Consider a Hamiltonian with the Sz operator: H = \ldots - \mus Bz Sz \ldots Here, the term is proportional to Sz, where \mus is the magnetic moment and B_z is the magnetic field along the z-axis.

• Eigenvectors are functions of r, \theta, \phi plus a column vector. The eigenfunction behaves similarly to a separation of variables because the Hamiltonian and the matrix commute. Thus, the spatial and spin parts of the wave function can be treated separately.

Non-Commutation of Components

• The components of the spin operator do not commute, which means that the order in which they are applied matters:

  • [Sz, Sy] = i \hbar S_x

  • [Sy, Sx] = i \hbar S_z

  • [Sx, Sy] = i \hbar S_z

• Because of this non-commutation, the eigenfunctions of Sx are not the same as those of Sz. If the system is locked into an eigenstate of Sz, a measurement of Sx will yield a distribution of possible values.

• The state \begin{bmatrix} 1 \ 0 \end{bmatrix} is not an eigenfunction of Sx. The eigenfunctions of the Sx matrix operator are linear combinations of the S_z eigenfunctions.

• Problem: Finding the eigenfunctions of Sx involves solving the equation: Sx ( a \begin{bmatrix} 1 \ 0 \end{bmatrix} + b \begin{bmatrix} 0 \ 1 \end{bmatrix}) = constant ( a \begin{bmatrix} 1 \ 0 \end{bmatrix} + b \begin{bmatrix} 0 \ 1 \end{bmatrix})

  • The constants will be +\frac{\hbar}{2} and -\frac{\hbar}{2}.

• When S_x operates on the state, the matrix multiplication yields: \frac{\hbar}{2} \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} a \ b \end{bmatrix} = \frac{\hbar}{2} \begin{bmatrix} b \ a \end{bmatrix}

Perturbation Theory

• Normalization condition: a^2 + b^2 = 1. With two equations and three unknowns, the normalization condition is essential to find a unique solution.

• The solution leads to: S{xj}^2 = \frac{\hbar^2}{4}. Thus, S{xj} = \pm \frac{\hbar}{2}.

• Therefore, the eigenvectors for the S_x operator are \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \ 1 \end{bmatrix}.

Spin Orbit Coupling

• Spin-orbit coupling refers to the interaction between an electron's intrinsic spin angular momentum (or its associated magnetic dipole moment vector) and the magnetic field generated by the orbiting proton. This interaction occurs internally within the atom and does not involve an external magnetic field.

• This interaction introduces a new term in the Hamiltonian, which describes this coupling. The new Hamiltonian is given by: H{new} = H{old} + H_{spin-orbit}, where the old Hamiltonian includes terms for kinetic energy, potential energy, and any external fields. This spin-orbit coupling leads to a further splitting of energy levels, although the magnitude of this splitting is typically small compared to the other terms in the Hamiltonian.

Magnetic Field

• To understand the spin-orbit interaction, imagine the proton orbiting the electron. The magnetic field due to the proton can be found using the Biot-Savart law:
  dB = \frac{\mu0}{4 \pi} \frac{Idl \times r}{r^3}, where dB is the infinitesimal magnetic field, \mu0 is the permeability of free space, I is the current, dl is the infinitesimal length vector, and r is the distance from the current element to the point where the field is being calculated.

• For a single charge, the magnetic field is given by: B = \frac{\mu_0}{4 \pi} \frac{q v \times r}{r^3}, where q is the charge, and v is the velocity of the proton. The direction of the field is given by the right-hand rule.

• Since v is perpendicular to r, the magnitude of the magnetic field can be simplified to: Bz = \frac{\mu0 q v r}{4 \pi r^3} sin(90^\circ) .

• The velocity of the proton as seen by the electron is equal to the negative of the velocity of the electron as seen by the proton: v{pe} = -v{ep}. Therefore,
  Bz = \frac{\mu0 q (-v_{electron})}{4 \pi r^2}.

• The magnetic field can be related to the orbital angular momentum of the electron. Given L = mr \times v{electron}, the electron's orbital angular momentum is related to the magnetic field created by the proton: Bz = \frac{\mu_0 q L}{4 \pi m r^3}.

New Hamiltonian

• The new Hamiltonian term is H' = -\mue \cdot Bp, where Bp is the magnetic field due to the proton. The magnetic dipole moment of the electron is \mue = \frac{q}{m} S_z, with an approximate Lande g-factor of 2.

• Thus, Bz = \frac{\mu0 q}{4 \pi mr^3} L_z.

• The new term consists of a collection of constants with a \frac{1}{r^3} term multiplied by L \cdot S. Since the \frac{1}{r^3} term does not commute with the original Hamiltonians, this new term also does not commute.

• However, because the new term is small, perturbation theory can be used to approximate the energy change. The first-order energy correction is given by:
  \Delta E = \langle \psi{old} | H{spin-orbit} | \psi_{old} \rangle . This involves taking the new operator and sandwiching it between the eigenfunctions of the old operator.

• This change in energy provides an estimate, ignoring the non-commutation of the \frac{1}{r^3} term with the original Hamiltonians. According to Perturbation Theory, adding a small term to the Hamiltonian allows us to determine how it changes the energy.

• Spin orbit coupling changes the energy by a tiny amount, leading to a small splitting of energy levels. This change in energy is either positive or negative due to the Lz and Sz terms in the matrix elements.

• Consequently, we observe splitting, but the effect is relatively small because this term is smaller than the other terms in the Hamiltonian.

Exchange Symmetry

• When considering a system with two particles, if their coordinates are not intermingled in the Hamiltonian, the states can be solved through separation of variables. This means that one particle operates in coordinates r1, \theta1, \phi1, while the other operates in r2, \theta2, \phi2.

• If the particles are interacting, the Hamiltonian may not be separable, leading to more complex solutions.

Two Identical Particles

• Now, consider two non-interacting particles confined within a box. The eigenfunction for this system is expressed as:
  \psi{12a} = C sin(\frac{n \pi x1}{L}) sin(\frac{m \pi x_2}{L}), where C is a normalization constant, n and m are quantum numbers, and L is the length of the box.

• Another state with the same energy is:
  \psi{12b} = C sin(\frac{n \pi x2}{L}) sin(\frac{m \pi x_1}{L}).

• The total energy for both states is E = \frac{\pi^2}{2mL^2} (n^2 + m^2) , indicating a degeneracy of two since both states have the same energy.

• When measuring the particle to be in this energy state, it could be in some linear combination of these two states, reflecting the uncertainty in which particle is in which quantum state.

• Particles with half-integer spin (such as 1/2, 3/2, 5/2, etc.) cannot occupy the same quantum state. This is known as the Pauli exclusion principle.

Pauli Exclusion Principle

• The Pauli exclusion principle asserts that particles with spin equal to one half, three halves, five halves, etc., cannot occupy the same state as another particle of the same type. This principle has profound implications for the structure of atoms and the behavior of matter.

• Wave functions for such systems must be antisymmetric, meaning that when two particles are exchanged, the wave function changes sign. These wave functions look like this, preventing the two particles from being in the same state:
  \psi = C [sin(\frac{n \pi x1}{L}) sin(\frac{m \pi x2}{L}) - sin(\frac{m \pi x1}{L}) sin(\frac{n \pi x2}{L})].

• This is called an antisymmetric wave function of two indistinguishable particles. It ensures that the wave function changes sign upon the exchange of particles.

• The anti-symmetric requirement prevents particles from occupying the same state. If two particles were in the same state, n and m would be identical, resulting in the wave function becoming zero.

Atoms and Electron Shells

• The Pauli exclusion principle dictates that when adding electrons to atoms, they cannot all occupy the same quantum state. This helps explain the large size of atoms as more electrons are added. According to the principle, each electron must occupy a unique quantum state, defined by a unique set of quantum numbers.

• In high school chemistry, you learned about electron shells (e.g., the k-shell, the l-shell), which represent electrons occupying different states. Electrons can't all be in n=1 states. There's an n=1 spin up and an n=1 spin down with l=0.

• The first electron occupies the state |1, 0, 0, \frac{1}{2}, +\frac{1}{2}\rangle (1s1). The label 1s1 refers to the electron with principle quantum number 1, orbital angular momentum 0, and spin up.

• The next electron occupies the state |1, 0, 0, \frac{1}{2}, -\frac{1}{2}\rangle (1s2). The label 1s2 refers to the electron with principle quantum number 1, orbital angular momentum 0, and spin down.

• A third electron must go to the n=2 state because there is no room left in n=1. Therefore, the third electron occupies the state |2, 0, 0, \frac{1}{2}, +\frac{1}{2}\rangle.

• There are eight possible states for n=2. This helps us understand the periodic table, where the layout of the table is based on the different shells occupied by the electrons.

Indistinguishable Particles

• When conducting experiments involving two indistinguishable particles, it is fundamentally impossible to determine which particle ends up in which final state after an interaction. This indistinguishability is a key concept in quantum mechanics.

• In a scattering experiment between two electrons, after a collision, there is no experimental way to determine which electron resides in which state. The uncertainty arises because the properties of identical particles are exactly the same, making them fundamentally indistinguishable.

• However, the particle must be truly indistinguishable within the system for this principle to hold. The indistinguishability is not valid if the two particles are not involved in the same system or if they can be distinguished by some external means.

• For example, there is no interaction between one electron and one lithium atom on one side of the room, and another electron in a different lithium atom on the other side of the room. Therefore, the Pauli exclusion principle would not apply in this