En mecánica cuántica , un nivel de energía está degenerado si corresponde a dos o más estados mensurables diferentes de un sistema cuántico . A la inversa, se dice que dos o más estados diferentes de un sistema de mecánica cuántica son degenerados si dan el mismo valor de energía en la medición. El número de estados diferentes correspondientes a un nivel de energía particular se conoce como el grado de degeneración del nivel. Está representado matemáticamente por el hamiltoniano para el sistema que tiene más de un autoestado linealmente independiente con el mismo autovalor de energía . [1] : pág. 48Cuando este es el caso, la energía por sí sola no es suficiente para caracterizar en qué estado se encuentra el sistema, y se necesitan otros números cuánticos para caracterizar el estado exacto cuando se desea la distinción. En la mecánica clásica , esto se puede entender en términos de diferentes trayectorias posibles correspondientes a la misma energía.
La degeneración juega un papel fundamental en la mecánica estadística cuántica . Para un sistema de N- partículas en tres dimensiones, un solo nivel de energía puede corresponder a varias funciones de onda o estados de energía diferentes. Estos estados degenerados en el mismo nivel son igualmente probables de ser llenados. El número de tales estados da la degeneración de un nivel de energía particular.
Matemáticas
Los posibles estados de un sistema de mecánica cuántica pueden tratarse matemáticamente como vectores abstractos en un espacio de Hilbert complejo y separable , mientras que los observables pueden representarse mediante operadores hermitianos lineales que actúan sobre ellos. Seleccionando una base adecuada , se pueden determinar los componentes de estos vectores y los elementos de la matriz de los operadores en esa base. Si A es una matriz N × N , X un vector distinto de cero y λ es un escalar , tal que, entonces se dice que el escalar λ es un valor propio de A y que el vector X es el vector propio correspondiente a λ . Junto con el vector cero, el conjunto de todos los autovectores correspondientes a un autovalor dado λ forma un subespacio de ℂ n , que se denomina autoespacio de λ . Un valor propio λ que corresponde a dos o más vectores propios linealmente independientes diferentes se dice que es degenerado , es decir, y , dónde y son vectores propios linealmente independientes. La dimensión del espacio propio correspondiente a ese valor propio se conoce como su grado de degeneración , que puede ser finita o infinita. Se dice que un valor propio no es degenerado si su espacio propio es unidimensional.
Los valores propios de las matrices que representan observables físicos en la mecánica cuántica dan los valores medibles de estos observables, mientras que los estados propios correspondientes a estos valores propios dan los estados posibles en los que se puede encontrar el sistema, después de la medición. Los valores medibles de la energía de un sistema cuántico están dados por los valores propios del operador hamiltoniano, mientras que sus estados propios dan los posibles estados de energía del sistema. Se dice que un valor de energía está degenerado si existen al menos dos estados de energía linealmente independientes asociados con él. Además, cualquier combinación lineal de dos o más estados propios degenerados es también un estado propio del operador hamiltoniano correspondiente al mismo valor propio de energía. Esto se deriva claramente del hecho de que el autoespacio del valor de energía autovalor λ es un subespacio (siendo el núcleo del hamiltoniano menos λ multiplicado por la identidad), por lo tanto, está cerrado bajo combinaciones lineales.
Prueba del teorema anterior. [2] : pág. 52 Si representa el operador hamiltoniano y y son dos estados propios correspondientes al mismo valor propio E , entonces Dejar , dónde y son constantes complejas (en general), sea cualquier combinación lineal de and . Then,
which shows that is an eigenstate of with the same eigenvalue E.
Efecto de la degeneración en la medición de la energía.
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 gn, the eigenstates associated with it form a vector subspace of dimension gn. In such a case, several final states can be possibly associated with the same result , all of which are linear combinations of the gn 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.
Degeneración en diferentes dimensiones
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 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 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 .[3] In particular, is bounded below in this criterion.
Proof of the above theorem. Considering a one-dimensional quantum system in a potential with degenerate states and corresponding to the same energy eigenvalue , writing the time-independent Schrödinger equation for the system: Multiplying the first equation by and the second by and subtracting one from the other, we get:
Integrating both sides
In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: where is, in general, a complex constant. For bound state eigenfunctions (which tend to zero as ), and assuming and satisfy the condition given above, it can be shown[3] that also the first derivative of the wave function approaches zero in the limit , so that the above constant is zero and we have no degeneracy.
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, 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 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 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 (nx = 7, ny = 1), (nx = 1, ny = 7) and (nx = ny = 5) all have and constitute a degenerate set.
Degrees of degeneracy of different energy levels for a particle in a square box:
Degeneracy | |||
---|---|---|---|
1 | 1 | 2 | 1 |
2 1 | 1 2 | 5 5 | 2 |
2 | 2 | 8 | 1 |
3 1 | 1 3 | 10 10 | 2 |
3 2 | 2 3 | 13 13 | 2 |
4 1 | 1 4 | 17 17 | 2 |
3 | 3 | 18 | 1 |
... | ... | ... | ... |
7 5 1 | 1 5 7 | 50 50 50 | 3 |
... | ... | ... | ... |
8 7 4 1 | 1 4 7 8 | 65 65 65 65 | 4 |
... | ... | ... | ... |
9 7 6 2 | 2 6 7 9 | 85 85 85 85 | 4 |
... | ... | ... | ... |
11 10 5 2 | 2 5 10 11 | 125 125 125 125 | 4 |
... | ... | ... | ... |
14 10 2 | 2 10 14 | 200 200 200 | 3 |
... | ... | ... | ... |
17 13 7 | 7 13 17 | 338 338 338 | 3 |
Particle in a cubical 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.
Encontrar una base propia única en caso de degeneración
If two operators 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 {a,b}, then and are said to form a complete set of commuting observables. 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 .
Degeneración y simetría
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 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 , since S is unitary. If the Hamiltonian remains unchanged under the transformation operation S, 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 E. If the two states and are linearly independent (i.e. physically distinct), they are therefore degenerate.
In cases where S 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 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.
Tipos de degeneración
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.[4] 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 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.
Example: 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.
Ejemplos de
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 , 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 n. For a given n, all the states corresponding to have the same energy and are degenerate. Similarly for given values of n and l, the , states with are degenerate. The degree of degeneracy of the energy level En is therefore :, which is doubled if the spin degeneracy is included.[1]:p. 267f
The degeneracy with respect to is an essential degeneracy which is present for any central potential, 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.[1]:p. 267f
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 acting on it is rotationally invariant, i.e. :
where is the angular frequency 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 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 degerneracy of the state,
As shown, only the ground state where is non-degenerate (ie, has a degerancy of ).
Eliminar la degeneración
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.
Lifting of degeneracy by first-order degenerate perturbation theory. Consider an unperturbed Hamiltonian and perturbation , so that the perturbed Hamiltonian The perturbed eigenstate, for no degeneracy, is given by-
The perturbed energy eigenket as well as higher order energy shifts diverge when , i.e., in the presence of degeneracy in energy levels. Assuming possesses N degenerate eigenstates with the same energy eigenvalue E, and also in general some non-degenerate eigenstates. A perturbed eigenstate can be written as a linear expansion in the unperturbed degenerate eigenstates as-
where refer to the perturbed energy eigenvalues. Since is a degenerate eigenvalue of ,
Premultiplying by another unperturbed degenerate eigenket gives-
This is an eigenvalue problem, and writing , we have-
The N eigenvalues obtained by solving this equation give the shifts in the degenerate energy level due to the applied perturbation, while the eigenvectors give the perturbed states in the unperturbed degenerate basis . To choose the good eigenstates from the beginning, it is useful to find an operator which commutes with the original Hamiltonian and has simultaneous eigenstates with it.
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 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, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.
- Ammonia molecule, where the Nitrogen atom can be either above or below the plane defined by the three Hydrogen atoms.
- H+2 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.
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 basis where the perturbation Hamiltonian is diagonal, is given by
where is the Bohr radius. The total fine-structure energy shift is given by
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 of the atom with the applied field is known as the Zeeman effect.
Taking into consideration the orbital and spin angular momenta, and , respectively, of a single electron in the Hydrogen atom, the perturbation Hamiltonian is given by
where and . Thus,
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 and are not separately conserved. The good quantum numbers are n, l, j and mj, and in this basis, the first order energy correction can be shown to be given by
- , where
is called the Bohr Magneton.Thus, depending on the value of , each degenerate energy level splits into several levels.
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 in the basis. It can be shown by the selection rules that when and .
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 and , both corresponding to n = 2, is given by .
Ver también
- Density of states
Referencias
- ^ a b c Merzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). New York: John Wiley. ISBN 0471887021.CS1 maint: uses authors parameter (link)
- ^ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice Hall. p. 52. ISBN 0-205-12770-3.CS1 maint: uses authors parameter (link)
- ^ a b Messiah, Albert (1967). Quantum mechanics (3rd ed.). Amsterdam, NLD: North-Holland. pp. 98–106. ISBN 0471887021.CS1 maint: uses authors parameter (link)
- ^ McIntosh, Harold V. (1959). "On Accidental Degeneracy in Classical and Quantum Mechanics" (PDF). American Journal of Physics. American Association of Physics Teachers (AAPT). 27 (9): 620–625. doi:10.1119/1.1934944. ISSN 0002-9505.
Otras lecturas
- Cohen-Tannoudji, Claude; Diu, Bernard & Laloë, Franck. Quantum Mechanics. 1. Hermann. ISBN 9782705683924.CS1 maint: uses authors parameter (link)[full citation needed]
- Shankar, Ramamurti (2013). Principles of Quantum Mechanics. Springer. ISBN 9781461576754.CS1 maint: uses authors parameter (link)[full citation needed]
- Larson, Ron; Falvo, David C. (30 March 2009). Elementary Linear Algebra, Enhanced Edition. Cengage Learning. pp. 8–. ISBN 978-1-305-17240-1.
- Hobson; Riley. Mathematical Methods For Physics And Engineering (Clpe) 2Ed. Cambridge University Press. ISBN 978-0-521-61296-8.
- Hemmer (2005). Kvantemekanikk: P.C. Hemmer. Tapir akademisk forlag. Tillegg 3: supplement to sections 3.1, 3.3, and 3.5. ISBN 978-82-519-2028-5.
- Quantum degeneracy in two dimensional systems, Debnarayan Jana, Dept. of Physics, University College of Science and Technology
- Al-Hashimi, Munir (2008). Accidental Symmetry in Quantum Physics.