Quantum Mechanics

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Section 3.2 Quantum Mechanics

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Objectives

You should be able to:

Explain why the states of a quantum mechanical system can be labeled using irreducible representations of the symmetry group, and how the degeneracy of states is computed by looking at the dimensions of the irreducible representations.
Use representation theory to determine the form of the states in the case of a cyclic periodic potential (Bloch's theorem).

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In this section we give a very brief introduction to the power of group and representation theory in quantum mechanics.

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Subsection 3.2.1 Schrodinger equation

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The starting point of quantum mechanics is Schrodinger equation. The state of a physical system in quantum mechanics is determined by a wave-function

$\Psi\text{,}$ which is a solution of a differential equation known as the Schrodinger equation of the system:

$\begin{equation} i \hbar \frac{\partial}{\partial t} \Psi = H \Psi,\label{equation-schrodinger}\tag{3.2.1} \end{equation}$

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where

$H$ is the Hamiltonian, which is a linear differential operator. It can be seen as a quantization of the Hamiltonian of classical mechanics, as we will see later on.

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Remark 3.2.1.

There is something important here that is not often mentioned: quantum mechanics is in many ways easier than classical mechanics. That is because in quantum mechanics, we are solving a linear differential equation to determine the wave-function $\Psi\text{.}$ So the superposition principle applies; given two solutions $\Psi_1$ and $\Psi_2$ of the Schrodinger equation, any linear combination of those is also a solution. So we can really think of the vector space of solutions (the Hilbert space of states), and think of the Hamiltonian as an operator on that space (if it is finite-dimensional, we can choose a basis and represent it as a matrix).

On the contrary, in classical mechanics, to solve a problem one needs to solve the equations of motion for, say, the position function $x(t)\text{.}$ Newton's law can be rewritten as

$\begin{equation*} m \frac{d^2}{dt^2} \vec{x}(t) = - \nabla V(\vec{x}). \end{equation*}$

These equations of motion for $\vec{x}(t)$ are generally non-linear, except for the case of harmonic motion, i.e. masses connected by springs, in which case $\nabla V(\vec{x})$ is linear in $\vec{x}\text{.}$ So in general, the superposition principle does not apply in classical mechanics, and linear combinations of solutions are not solutions of the equations of motion.

This is why group and representation theory are much more useful and powerful in quantum mechanics than in classical mechanics.

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To solve Schrodinger equation we do a separation of variables. We write

$\Psi = \psi e^{-i E t/\hbar}\text{,}$ where

$\psi$ does not depend on time. Substituting back in (3.2.1), we get:

$\begin{equation*} H \psi = E \psi, \end{equation*}$

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which is an eigenvalue problem for the Hamiltonian of the system.

$E$ here is the energy of the system. This is the key equation that we need to solve.

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Subsection 3.2.2 The symmetry group of the Hamiltonian

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The power of group and representation theory is that we will be allowed to characterize the form of the space of solutions for a given quantum mechanical problem without ever solving the eigenvalue problem. All that we will use are the symmetry properties of the system.

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Suppose that

$R$ is another operator acting on the space of states. We know how an operator transforms: it transforms by a similar transformation. Thus, the Hamiltonian transforms as

$\begin{equation*} H' = R^{-1} H R. \end{equation*}$

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We say that the Hamiltonian is invariant under

$R$ if

$H' = H\text{,}$ that is,

$\begin{equation*} H = R^{-1} H R, \end{equation*}$

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or, equivalently,

$\begin{equation*} R H = H R. \end{equation*}$

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We can look at all such operators

$R$ that commute with

$H\text{:}$ those form a group

$G\text{,}$ which we call the symmetry group of the Hamiltonian.

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More precisely, we could start with the symmetry group

$G\text{,}$ realized in terms of properties of the Hamiltonian (for instance some explicit symmetry of the system, such as periodicity). Then the operators

$R$ would form a representation of the group

$G\text{,}$ acting on the space of states of the system.

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Subsection 3.2.3 Irreducible representations and degeneracy of states

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Now, the key is to realize that two eigenfunctions that are mapped into each other by a symmetry transformation

$R$ must have the same energy eigenvalue

$E\text{.}$ Indeed, if

$H \psi = E \psi\text{,}$ then

$\begin{equation*} H (R \psi) = R (H \psi) = R (E \psi) = E (R \psi). \end{equation*}$

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So what this means is that we can group the eigenfunctions of the systems into subspaces that are invariant under the action of the symmetry group; all eigenfunctions in each of these subspaces share the same eigenvalue.

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This statement follows neatly from representation theory. If

$R$ is an irreducible presentation, then, since

$H R = R H\text{,}$ by Schur's lemma we must have

$H = E I$ for some number

$E\text{.}$ What this means is that all states have the same energy eivenvalue. However, in general

$R$ is not irreducible. But one can choose a basis for the space of states wuch that

$R$ decomposes as direct sum of irreducible representations

$R = \bigoplus_\alpha R^{(\alpha)}\text{,}$ and by Schur's lemma, it follows that with respect to this choice of basis the Hamiltonian also decomposes as a direct sum of diagonal matrices

$H = \bigoplus_\alpha E^{(\alpha)} I\text{,}$ where

$I$ is the identity matrix of the size of the corresponding irreducible representation

$R^{(\alpha)}\text{.}$

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To summarize, what this means is that we can choose a basis for the space of states, composed of eigenfunctions of the Hamiltonian, in which the state space decomposes as a direct sum of subspaces that transform according to irreducible representations of the symmetry group. All states in each such subspace are eigenfunctions of the Hamiltonian with the same energy eigenvalue.

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This leads to the very important physical concept of degeneracy.

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Definition 3.2.2. Degeneracy of states.

We say that two eigenfunctions of the Hamiltonian are degenerate if they share the same energy eigenvalue.

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At first, in the history of physics, when eigenstates that shared the same energy eigenvalues were discovered, it was very puzzling to physicists. What should states have the same energy eigenvalue? When you diagonalize a generic matrix, you expect distinct eigenvalues.

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But from the viewpoint of the representation theory, degeneracy is expected, and in fact can be constrained using representation theory. Indeed, we know that the “degree of degeneracy” of an eigenfunction (that is, the number of degenerate states) is at least as large as the dimension of the irreducible representation that it belongs to. It can be larger if two eigenspaces corresponding to different irreducible representations have the same energy eigenvalue; we call such degeneracy “accidental”, since it is not determined by symmetry. However, most degeneracies are determined by symmetry.

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This provides a beautiful interplay between mathematics and physics. On the one hand, if we know the group of symmetry of a system, we can determine the possible degeneracies, simply by enumerating its irreducible representations and their dimensions. On the other hand, if we observe experimentally that say 8 states are degenerate (as was observed in the 50s when eight baryons were observed with approximately the same mass), then we can use this information to restrict the possible groups of symmetries: assuming that the degeneracy is not accidental, the group of symmetries must be such that it has an 8-dimensional irreducible representation.

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The upshot of this discussion could be summarized as follows:

To each energy eigenvalue we can associate a corresponding irreducible representation of the symmetry group of the Hamiltonian. The degeneracy of the eigenvalue is the dimension of the irreducible representation (it could be larger if there is accidental degeneracy).
In the basis for the space of states given by eigenfunctions of the Hamiltonian, the state space decomposes as a direct sum of subspaces that transform according to irreducible representations of the symmetry group. All eigenfunctions in a given subspace share the same energy eigenvalue. Thus, we can label the eigenfunctions of the system using the irreducible representations of the symmetry group, which we call in physics the quantum numbers.

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Subsection 3.2.4 Examples

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We will look at two simple examples of the construction. But most relevant examples to quantum mechanics involve continuous groups and their representations, thus for this we will need to wait until later.

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Example 3.2.3. Parity.

Let us start with a very simple one-dimensional example. The Hamiltonian of a one-dimensional system takes the form $H = - \frac{1}{2} \frac{d^2}{dx^2} + V(x)$ for some potential function $V(x)\text{.}$ Now suppose that $V(-x)=V(x)\text{.}$ Then the mapping $x \mapsto -x$ leaves the Hamiltonian $H$ invariant. This symmetry group is the order two group $\mathbb{Z}_2 = \{e,a\}\text{.}$

This means that we can label the eigenfunctions of the system according to irreducible representations of the symmetry group. Here, $\mathbb{Z}_2$ has only two one-dimensional irreducible representations: the trivial representation (given by $T(e)=T(a) = 1$), and the parity representation (given by $P(e)=1\text{,}$ $P(a)=-1$). Those act on states of the system (eigenfunctions). States that transform according to the trivial representation satisfy:

\begin{equation*} \psi(-x) = T(a) \psi(x) = \psi(x), \end{equation*}

that is, they are even. States that transform according to the parity representation transform as:

\begin{equation*} \psi(-x) = S(a) \psi(x) = - \psi(x), \end{equation*}

that is, they are odd. The upshot of the representation theory is that we can label the states according to whether they are even or odd.

We also know that there is no degeneracy here (barring accidental degeneracy), since all irreducible representations of $\mathbb{Z}_2$ are one-dimensional.

This example is of course almost trivial: the goal was just to get familiar with how to use the language of representation theory to understand states of a quantum mechanical system.

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Example 3.2.4. Bloch's theorem.

The second example is more interesting, and in fact is of fundamental importance in solid state physics. We stick with a one-dimensional system, but we consider a potential $V(x)$ that is periodic: $V(x+a) = V(x)$ for some constant $a\text{.}$ We consider the mapping $T: x \to x+a\text{.}$ This generates a discrete, infinite, abelian group $G\text{.}$

Since $G$ is abelian, it only has one-dimensional irreducible representations. So as before, there is no degeneracy of states. We can label states using irreducible representations. Those representations are unitary. So for a one-dimensional irreducible representation $S_k$ we get the transformation:

\begin{equation*} \psi(x+a) = S_k \psi(x) = e^{i k a} \psi(x), \end{equation*}

for some real number $k\text{.}$ Since $e^{i k a} = e^{i k a + i 2\pi}\text{,}$ the representations are indexed by a real number $k$ such that

\begin{equation*} - \frac{\pi}{a} \leq k \lt \frac{\pi}{a}. \end{equation*}

So the states of the system can be labeled by such a real number. This is an example of a Brillouin zone.

We can look at the special case of a one-dimensional crystal (note that it can be easily generalized to more than one dimension) with a finite number of sites, say $N\text{.}$ We can impose a periodic boundary condition, meaning that if we translate from the $N$th site, we go back to the first one. Thus the group of symmetry is generated by the translation $T\text{,}$ but with the condition $T^n = e\text{.}$ This group is the cyclic group $\mathbb{Z}_N\text{.}$

The abelian group $\mathbb{Z}_N$ has $N$ one-dimensional irreducible representations $S_k\text{.}$ Those are given by representing the generator $T$ by the roots of unity $S_k(T) = w^k\text{,}$ for $k=0,1,\ldots,N-1\text{,}$ where $w = e^{2 \pi i /N}\text{.}$ So we can label the states according to how they transform. For instance, a state $\psi_k$ that transforms in the $k$th irreducible representation would transform as

\begin{equation} \psi_k(x+a) = S_k(T) \psi_k(x) = e^{2 \pi i k / N} \psi_k(x).\label{equation-trans-bloch}\tag{3.2.2} \end{equation}

Now let us write the state $\psi_k(x)$ as

\begin{equation} \psi_k(x) = e^{i \phi_k(x)} u_k(x),\label{equation-phi-bloch}\tag{3.2.3} \end{equation}

where $u_k(x+a) = u_k(x)$ and $\phi_k(x)$ is some phase factor. Because of (3.2.2), we must have:

\begin{equation*} \psi_k(x+a) = e^{i \phi_k(x+a)} u(x) = e^{2 \pi i k / N} \psi_k(x) = e^{2 \pi i k / N + i \phi_k(x)} u(x) . \end{equation*}

Thus we must have

\begin{equation} \phi_k(x+a) = \frac{2 \pi k}{N} + \phi_k(x),\label{equation-phase-bloch}\tag{3.2.4} \end{equation}

and, iterating, $\phi_k(x+m a) = \frac{2 \pi k m}{N} + \phi_k(x)\text{.}$ Thus $\phi_k$ is a linear function of $m\text{,}$ and hence of $x + m a\text{.}$ Therefore we can write $\phi_k(x) = A x + B$ for some $A$ and $B\text{.}$ Substituting back in (3.2.4), we get:

\begin{equation*} A = \frac{2 \pi k}{N a}. \end{equation*}

Thus $\phi_k(x) = \frac{2 \pi k}{N a} x + B\text{.}$ Subsituting back in (3.2.3), and redefining $u_k(x)$ to include in it the arbitrary constant $B\text{,}$ we get:

\begin{equation*} \psi_k(x) = e^{\frac{2 \pi i k x}{N a}} u_k(x). \end{equation*}

This is known as a Bloch wave: it is a periodic function $u_k(x)$ multiplied by a plane wave. Representation theory tells us that the quantum mechanical states of a periodic lattice have this form. This is pretty much the statement of Bloch's theorem: that a basis of wavefunctions is given by Bloch waves, and that these wavefunctions are energy eigenstates of the system.

The amazing thing is that this result follows directly from representation theory, without ever solving the Schrodinger equation! All that we used is the periodic symmetry of the lattice. The particular form of the potential would then be needed to solve for the periodic functions $u_k(x)$ and identity the particular energy eigenvalues corresponding to the eigenstates of the system.