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In quantum mechanics, an energy level is said to be degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.
Contents
- Mathematics
- Effect of degeneracy on the measurement of energy
- Degeneracy in different dimensions
- Degeneracy in one dimension
- Degeneracy in two dimensional quantum systems
- Particle in a rectangular plane
- Particle in a square box
- Finding a unique eigenbasis in case of degeneracy
- Choosing a complete set of commuting observables
- Degenerate energy eigenstates and the parity operator
- Degeneracy and symmetry
- Symmetry group of the Hamiltonian
- Types of degeneracy
- Systematic or essential degeneracy
- Accidental degeneracy
- The Coulomb and Harmonic Oscillator potentials
- Particle in a constant magnetic field
- The hydrogen atom
- Isotropic three dimensional harmonic oscillator
- Removing degeneracy
- Physical examples of removal of degeneracy by a perturbation
- Symmetry breaking in two level systems
- Fine structure splitting
- Zeeman effect
- Stark effect
- References
Degeneracy plays a fundamental role in quantum statistical mechanics. For an N-particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level are all equally probable of being filled. The number of such states gives the degeneracy of a particular energy level.
Mathematics
The possible states of a quantum mechanical system may be treated mathematically as abstract vectors in a separable, complex Hilbert space, while the observables may be represented by linear Hermitian operators acting upon them. By selecting a suitable basis, the components of these vectors and the matrix elements of the operators in that basis may be determined. If A is a N × N matrix, X a non-zero vector, and λ is a scalar, such that
Effect of degeneracy on the measurement of energy
In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to each energy eigenvalue. However, if the Hamiltonian
In this case, the probability that the energy value measured for a system in the state
Degeneracy in different dimensions
This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of one and two-dimensional systems aids the conceptual understanding of more complex systems.
Degeneracy in one dimension
In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function
Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy
Degeneracy in two-dimensional quantum systems
Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions. Real two-dimensional materials are made of monatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems.
Particle in a rectangular plane
Consider a free particle in a plane of dimensions
The permitted energy values are
The normalized wave function is
where
So, quantum numbers
For some commensurate ratios of the two lengths
Particle in a square box
In this case, the dimensions of the box
Since
Degrees of degeneracy of different energy levels for a particle in a square box
Finding a unique eigenbasis in case of degeneracy
If two operators
For two commuting observables A and B, one can construct an orthonormal basis of the state space with eigenvectors common to the two operators. However,
Choosing a complete set of commuting observables
If a given observable A is non-degenerate, there exists a unique basis formed by its eigenvectors. On the other hand, if one or several eigenvalues of
It follows that the eigenfunctions of the Hamiltonian of a quantum system with a common energy value must be labelled by giving some additional information, which can be done by choosing an operator that commutes with the Hamiltonian. These additional labels required naming of a unique energy eigenfunction and are usually related to the constants of motion of the system.
Degenerate energy eigenstates and the parity operator
The parity operator is defined by its action in the
The eigenvalues of P can be shown to be limited to
Now, an even operator
while an odd operator
Since the square of the momentum operator
Degeneracy and symmetry
The physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Studying the symmetry of a quantum system can, in some cases, enable us to find the energy levels and degeneracies without solving the Schrödinger equation.
Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Let us consider a symmetry operation associated with a unitary operator S. Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that
Now, if
where E is the corresponding energy eigenvalue.
which means that
In cases where S is characterized by a continuous parameter
Symmetry group of the Hamiltonian
The set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian. The commutators of the generators of this group determine the algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.
Types of degeneracy
Degeneracies in a quantum system can be systematic or accidental in nature.
Systematic or essential degeneracy
This is also called a geometrical or normal degeneracy and arises due to the presence of some kind of symmetry in the system under consideration, i.e. the invariance of the Hamiltonian under a certain operation, as described above. The representation obtained from a normal degeneracy is irreducible and the corresponding eigenfunctions form a basis for this representation.
Accidental degeneracy
It is a type of degeneracy resulting from some special features of the system or the functional form of the potential under consideration, and is related possibly to a hidden dynamical symmetry in the system. It also results in conserved quantities, which are often not easy to identify. Accidental symmetries lead to these additional degeneracies in the discrete energy spectrum. An accidental degeneracy can be due to the fact that the group of the Hamiltonian is not complete. These degeneracies are connected to the existence of bound orbits in classical Physics.
The Coulomb and Harmonic Oscillator potentials
For a particle in a central 1/r potential, the Laplace–Runge–Lenz vector is a conserved quantity resulting from an accidental degeneracy, in addition to the conservation of angular momentum due to rotational invariance.
For a particle moving on a cone under the influence of 1/r and r2 potentials, centred at the tip of the cone, the conserved quantities corresponding to accidental symmetry will be two components of an equivalent of the Runge-Lenz vector, in addition to one component of the angular momentum vector. These quantities generate SU(2) symmetry for both potentials.
Particle in a constant magnetic field
A particle moving under the influence of a constant magnetic field, undergoing cyclotron motion on a circular orbit is another important example of an accidental symmetry. The symmetry multiplets in this case are the Landau levels which are infinitely degenerate.
The hydrogen atom
In atomic physics, the bound states of an electron in a hydrogen atom show us useful examples of degeneracy. In this case, the Hamiltonian commutes with the total orbital angular momentum
The energy levels in the hydrogen atom depend only on the principal quantum number n. For a given n, all the states corresponding to
The degeneracy with respect to
Isotropic three-dimensional harmonic oscillator
It is a spinless particle of mass m moving in three-dimensional space, subject to a central force whose absolute value is proportional to the distance of the particle from the centre of force.
It is said to be isotropic since the potential
where
Since the state space of such a particle is the tensor product of the state spaces associated with the individual one-dimensional wave functions, the time-independent Schrödinger equation for such a system is given by-
So, the energy eigenvalues are
or,
where n is a non-negative integer. So, the energy levels are degenerate and the degree of degeneracy is equal to the number of different sets
which is equal to
Only the ground state is non-degenerate.
Removing degeneracy
The degeneracy in a quantum mechanical system may be removed if the underlying symmetry is broken by an external perturbation. This causes splitting in the degenerate energy levels. This is essentially a splitting of the original irreducible representations into lower-dimensional such representations of the perturbed system.
Mathematically, the splitting due to the application of a small perturbation potential can be calculated using time-independent degenerate perturbation theory. This is an approximation scheme that can be applied to find the solution to the eigenvalue equation for the Hamiltonian H of a quantum system with an applied perturbation, given the solution for the Hamiltonian H0 for the unperturbed system. It involves expanding the eigenvalues and eigenkets of the Hamiltonian H in a perturbation series. The degenerate eigenstates with a given energy eigenvalue form a vector subspace, but not every basis of eigenstates of this space is a good starting point for perturbation theory, because typically there would not be any eigenstates of the perturbed system near them. The correct basis to choose is one that diagonalizes the perturbation Hamiltonian within the degenerate subspace.
Physical examples of removal of degeneracy by a perturbation
Some important examples of physical situations where degenerate energy levels of a quantum system are split by the application of an external perturbation are given below.
Symmetry breaking in two-level systems
A two-level system essentially refers to a physical system having two states whose energies are close together and very different from those of the other states of the system. All calculations for such a system are performed on a two-dimensional subspace of the state space.
If the ground state of a physical system is two-fold degenerate, any coupling between the two corresponding states lowers the energy of the ground state of the system, and makes it more stable.
If
then the perturbed energies are
Examples of two-state systems in which the degeneracy in energy states is broken by the presence of off-diagonal terms in the Hamiltonian resulting from an internal interaction due to an inherent property of the system include:
Fine-structure splitting
The corrections to the Coulomb interaction between the electron and the proton in a Hydrogen atom due to relativistic motion and spin-orbit coupling result in breaking the degeneracy in energy levels for different values of l corresponding to a single principal quantum number n.
The perturbation Hamiltonian due to relativistic correction is given by
where
Now
where
The spin-orbit interaction refers to the interaction between the intrinsic magnetic moment of the electron with the magnetic field experienced by it due to the relative motion with the proton. The interaction Hamiltonian is
which may be written as
The first order energy correction in the
where
for
Zeeman effect
The splitting of the energy levels of an atom when placed in an external magnetic field because of the interaction of the magnetic moment
Taking into consideration the orbital and spin angular momenta,
where
Now, in case of the weak-field Zeeman effect, when the applied field is weak compared to the internal field, the spin-orbit coupling dominates and
In case of the strong-field Zeeman effect, when the applied field is strong enough, so that the orbital and spin angular momenta decouple, the good quantum numbers are now n,l,ml and ms. Here, Lz and Sz are conserved, so the perturbation Hamiltonian is given by-
assuming the magnetic field to be along the z-direction. So,
For each value of ml, there are two possible values of ms,
Stark effect
The splitting of the energy levels of an atom or molecule when subjected to an external electric field is known as the Stark effect.
For the hydrogen atom, the perturbation Hamiltonian is-
if the electric field is chosen along the z-direction.
The energy corrections due to the applied field are given by the expectation value of
The degeneracy is lifted only for certain states obeying the selection rules, in the first order. The first-order splitting in the energy levels for the degenerate states