In
quantum mechanics, an
energy level is 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
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
for the system having more than one
linearly independent eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
with the same energy
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
.
When this is the case, energy alone is not enough to characterize what state the system is in, and other
quantum numbers
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
are needed to characterize the exact state when distinction is desired. In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical m ...
, this can be understood in terms of different possible trajectories corresponding to the same energy.
Degeneracy plays a fundamental role in
quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
. For an -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 all have an equal probability 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
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
may be represented by
linear Hermitian operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to i ...
s acting upon them. By selecting a suitable
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
, the components of these vectors and the matrix elements of the operators in that basis may be determined.
If is a matrix, a non-zero
vector, and is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
, such that
, then the scalar is said to be an eigenvalue of and the vector is said to be the eigenvector corresponding to . Together with the zero vector, the set of all
eigenvectors corresponding to a given eigenvalue form a
subspace of , which is called the eigenspace of . An eigenvalue which corresponds to two or more different linearly independent eigenvectors is said to be degenerate, i.e.,
and
, where
and
are linearly independent eigenvectors. The
dimension of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracy, which can be finite or infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional.
The eigenvalues of the matrices representing physical
observables
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
in
quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon measurement. The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any
linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy eigenvalue. This clearly follows from the fact that the eigenspace of the energy value eigenvalue is a subspace (being the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
of the Hamiltonian minus times the identity), hence is closed under linear combinations.
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
has a degenerate eigenvalue
of degree g
n, the eigenstates associated with it form a
vector subspace of
dimension g
n. In such a case, several final states can be possibly associated with the same result
, all of which are linear combinations of the g
n orthonormal eigenvectors
.
In this case, the probability that the energy value measured for a system in the state
will yield the value
is given by the sum of the probabilities of finding the system in each of the states in this basis, i.e.
:
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
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
moving in a one-dimensional potential
, the
time-independent Schrödinger equation can be written as
:
Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy
at most, so that the degree of degeneracy never exceeds two. It can be proven that in one dimension, there are no degenerate
bound states
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
*B ...
for
normalizable wave functions. A sufficient condition on a piecewise continuous potential
and the energy
is the existence of two real numbers
with
such that
we have
.
In particular,
is bounded below in this criterion.
:
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 monoatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include
MOSFET
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET) is a type of field-effect transistor (FET), most commonly fabricated by the controlled oxidation of silicon. It has an insulated gate, the voltage of which d ...
, two-dimensional
superlattices of
Helium,
Neon
Neon is a chemical element with the symbol Ne and atomic number 10. It is a noble gas. Neon is a colorless, odorless, inert monatomic gas under standard conditions, with about two-thirds the density of air. It was discovered (along with krypton ...
,
Argon
Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice as abu ...
,
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
and
in a plane of impenetrable walls. The time-independent Schrödinger equation for this system with wave function
can be written as
:
The permitted energy values are
:
The normalized wave function is
:
where
So,
quantum numbers
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
and
are required to describe the energy eigenvalues and the lowest energy of the system is given by
:
For some commensurate ratios of the two lengths
and
, certain pairs of states are degenerate.
If
, where p and q are integers, the states
and
have the same energy and so are degenerate to each other.
Particle in a square box
In this case, the dimensions of the box
and the energy eigenvalues are given by
:
Since
and
can be interchanged without changing the energy, each energy level has a degeneracy of at least two when
and
are different. Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same. For example, the three states (n
x = 7, n
y = 1), (n
x = 1, n
y = 7) and (n
x = n
y = 5) all have
and constitute a degenerate set.
Degrees of degeneracy of different energy levels for a particle in a square box:
Particle in a cubic box
In this case, the dimensions of the box
and the energy eigenvalues depend on three quantum numbers.
:
Since
,
and
can be interchanged without changing the energy, each energy level has a degeneracy of at least three when the three quantum numbers are not all equal.
Finding a unique eigenbasis in case of degeneracy
If two
operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another s ...
s
and
commute, i.e.
, then for every eigenvector
of
,
is also an eigenvector of
with the same eigenvalue. However, if this eigenvalue, say
, is degenerate, it can be said that
belongs to the eigenspace
of
, which is said to be globally invariant under the action of
.
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,
is a degenerate eigenvalue of
, then it is an eigensubspace of
that is invariant under the action of
, so the
representation of
in the eigenbasis of
is not a diagonal but a
block diagonal matrix, i.e. the degenerate eigenvectors of
are not, in general, eigenvectors of
. However, it is always possible to choose, in every degenerate eigensubspace of
, a basis of eigenvectors common to
and
.
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
are degenerate, specifying an eigenvalue is not sufficient to characterize a basis vector. If, by choosing an observable
, which commutes with
, it is possible to construct an orthonormal basis of eigenvectors common to
and
, which is unique, for each of the possible pairs of eigenvalues , then
and
are said to form a
complete set of commuting observables In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of co ...
. However, if a unique set of eigenvectors can still not be specified, for at least one of the pairs of eigenvalues, a third observable
, which commutes with both
and
can be found such that the three form a complete set of commuting observables.
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
representation of changing r to −r, i.e.
:
The eigenvalues of P can be shown to be limited to
, which are both degenerate eigenvalues in an infinite-dimensional state space. An eigenvector of P with eigenvalue +1 is said to be even, while that with eigenvalue −1 is said to be odd.
Now, an even operator
is one that satisfies,
:
:
while an odd operator
is one that satisfies
:
Since the square of the momentum operator
is even, if the potential V(r) is even, the Hamiltonian
is said to be an even operator. In that case, if each of its eigenvalues are non-degenerate, each eigenvector is necessarily an eigenstate of P, and therefore it is possible to look for the eigenstates of
among even and odd states. However, if one of the energy eigenstates has no definite
parity, it can be asserted that the corresponding eigenvalue is degenerate, and
is an eigenvector of
with the same eigenvalue as
.
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, hence reducing effort.
Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Consider a
symmetry operation In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...
associated with a
unitary operator . Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a
similarity transformation generated by the operator , such that
, since is unitary. If the Hamiltonian remains unchanged under the transformation operation , we have
:
:
:
:
Now, if
is an energy eigenstate,
:
where E is the corresponding energy eigenvalue.
:
which means that
is also an energy eigenstate with the same eigenvalue . If the two states
and
are linearly independent (i.e. physically distinct), they are therefore degenerate.
In cases where is characterized by a continuous
parameter , all states of the form
have the same energy eigenvalue.
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
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
of the Hamiltonian. The
commutators of the
generators of this group determine the
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
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.
Examples: Coulomb and Harmonic Oscillator potentials
For a particle in a central potential, the
Laplace–Runge–Lenz vector
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For t ...
is a conserved quantity resulting from an accidental degeneracy, in addition to the conservation of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
due to rotational invariance.
For a particle moving on a cone under the influence of and 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.
Example: Particle in a constant magnetic field
A particle moving under the influence of a constant magnetic field, undergoing
cyclotron
A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Jan ...
motion on a circular orbit is another important example of an accidental symmetry. The symmetry
multiplets
In physics and particularly in particle physics, a multiplet is the state space for 'internal' degrees of freedom of a particle, that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as the ...
in this case are the
Landau levels which are infinitely degenerate.
Examples
The hydrogen atom
In
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned w ...
, the bound states of an electron in a
hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
show us useful examples of degeneracy. In this case, the Hamiltonian commutes with the total
orbital angular momentum , its component along the z-direction,
, total
spin angular momentum and its z-component
. The quantum numbers corresponding to these operators are
,
,
(always 1/2 for an electron) and
respectively.
The energy levels in the hydrogen atom depend only on the
principal quantum number . For a given , all the states corresponding to
have the same energy and are degenerate. Similarly for given values of and , the
, states with
are degenerate. The degree of degeneracy of the energy level E
n is therefore :
, which is doubled if the spin degeneracy is included.
[
The degeneracy with respect to is an essential degeneracy which is present for any ]central potential
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.
: \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat
where \vec F is the force, F is a vecto ...
, and arises from the absence of a preferred spatial direction. The degeneracy with respect to is often described as an accidental degeneracy, but it can be explained in terms of special symmetries of the Schrödinger equation which are only valid for the hydrogen atom in which the potential energy is given by Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
.[
]
Isotropic three-dimensional harmonic oscillator
It is a spinless particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
of mass m moving in three-dimensional space, subject to a central force
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.
: \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat
where \vec F is the force, F is a vecto ...
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 acting on it is rotationally invariant, i.e. :
where is the angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
given by .
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 satisfying
:
The degeneracy of the -th state can be found by considering the distribution of quanta across , and . Having 0 in gives possibilities for distribution across and . Having 1 quanta in gives possibilities across and and so on. This leads to the general result of and summing over all leads to the degeneracy of the -th state,
:
As shown, only the ground state where is non-degenerate (ie, has a degeneracy of ).
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
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a syst ...
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 and are the energy levels of the system, such that , and the perturbation is represented in the two-dimensional subspace as the following 2×2 matrix
:
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:
* Benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms ...
, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.
* Ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous wa ...
molecule, where the Nitrogen atom can be either above or below the plane defined by the three Hydrogen atoms.
* molecule, in which the electron may be localized around either of the two nuclei.
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 is the momentum operator and is the mass of the electron. The first-order relativistic energy correction in the basis is given by
:
Now
:
where is the fine structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between ele ...
.
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
: