The
quantum harmonic oscillator is the
quantum mechanicalQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
analogue of the
classical harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:where k is a positive constant....
. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable
equilibrium pointIn mathematics, the point is an equilibrium point for the differential equationif for all .Similarly, the point is an equilibrium point for the difference equationif for ....
. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution is known.
Hamiltonian and energy eigenstates
In the one-dimensional harmonic oscillator problem, a particle of mass
m is subject to a potential
V(
x) given by
where
ω is the
angular frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of the oscillator.
In classical mechanics,
is called the spring stiffness coefficient, force constant or spring constant, and
the
angular frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
.
The
HamiltonianIn quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of the particle is:
where
' is the positionSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of the boundless four-dimensional...
operator, and ' is the
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
operator, given by
The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents the potential energy in which it resides. In order to find the
energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
levels and the corresponding energy eigenstates, we must solve the time-independent
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
,
We can solve the differential equation in the coordinate basis, using a
spectral methodSpectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain partial differential equations , often involving the use of the Fast Fourier Transform...
. It turns out that there is a family of solutions,
The functions
Hn are the
Hermite polynomialsIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, where they give rise to the eigenstates of the quantum...
:
The corresponding energy levels are.
This energy spectrum is noteworthy for three reasons. Firstly, the energies are "quantized", and may only take the discrete half-integer multiples of . This is a feature of many quantum mechanical systems. In the following section on ladder operators, we will engage in a more detailed examination of this phenomenon. Secondly, the lowest achievable energy is not zero, but , which is called the "ground state energy" or
zero-point energyIn physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may have and is the energy of the ground state. The quantum mechanical system that encapsulates this energy is the zero-point field. The concept was first proposed by Albert Einstein and Otto...
. In the ground state, according to quantum mechanics, an oscillator performs null oscillations and its average kinetic energy is positive. It is not obvious that this is significant, because normally the zero of energy is not a physically meaningful quantity, only differences in energies. Nevertheless, the ground state energy has many implications, particularly in
quantum gravityQuantum gravity is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity and which reduces to...
. The final reason is that the energy levels are equally spaced, unlike the
Bohr modelIn atomic physics, the Bohr model, devised by Niels Bohr, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing...
or the
particle in a boxIn quantum mechanics, the particle in a box model describes a particle, which is free to move in a space surrounded by impenetrable barriers...
.
Note that the ground state probability density is concentrated at the origin. This means the particle spends most of its time at the bottom of the potential well, as we would expect for a state with little energy. As the energy increases, the probability density becomes concentrated at the "classical turning points", where the state's energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The
correspondence principleIn physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
is thus satisfied.
Ladder operator method
The spectral method solution, though straightforward, is rather tedious. The "
ladder operatorIn linear algebra , a raising or lowering operator is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator...
" method, due to
Paul DiracPaul Adrien Maurice Dirac, OM, FRS was a British theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
, allows us to extract the energy eigenvalues without directly solving the differential equation. Furthermore, it is readily generalizable to more complicated problems, notably in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
. Following this approach, we define the operators
a and its
adjointIn mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type = [x, By].Specifically, adjoint may mean:...
a†
The operator
a is not Hermitian since it and its adjoint
a† are not equal.
The operator
a and
a† have properties as below:
We can also define a number operator N which has the following property:
In deriving the form of
a†, we have used the fact that the operators x and p, which represent observables,
are Hermitian. These observable operators can be expressed as a linear combination of the ladder operators as
The
x and
p operators obey the following identity, known as the
canonical commutation relationIn physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:...
:
The square brackets in this equation are a commonly-used notational device, known as the
commutatorIn mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
, defined as.
Using the above, we can prove the identities
.
Now, let denote an energy eigenstate with energy
E. The inner product of any
ketBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in pure mathematics...
with itself must be non-negative, so
.
Expressing
a†a in terms of the Hamiltonian:
,
so that
E ≥
ħω2. Note that when is the zero ket (i.e. a ket with length zero), the inequality is saturated, so that . It is straightforward to check that there exists a state satisfying this condition; it is the ground (
n = 0) state given in the preceding section.
Using the above identities, we can now show that the commutation relations of
a and
a† with
H are:
Thus, provided is not the zero ket,
.
Similarly, we can show that
.
In other words,
a acts on an eigenstate of energy
E to produce, up to a multiplicative constant, another eigenstate of energy , and
a† acts on an eigenstate of energy
E to produce an eigenstate of energy . For this reason,
a is called a "lowering operator", and
a† a "raising operator". The two operators together are called
ladder operatorIn linear algebra , a raising or lowering operator is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator...
s. In quantum field theory,
a and
a† are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.
Given any energy eigenstate, we can act on it with the lowering operator,
a, to produce another eigenstate with -less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to
E = −∞. However, this would contradict our earlier requirement that . Therefore, there must be a ground-state energy eigenstate, which we label (not to be confused with the zero ket), such that
.
In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates. Furthermore, we have shown above that
Finally, by acting on with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates , such that
which matches the energy spectrum which we gave in the preceding section.
This method can also be used to quickly find the ground state wave function of the quantum harmonic oscillator.
Indeed becomes
so that
After normalization this leads to the following position space representation of the ground state wave function.
Natural length and energy scales
The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization. The result is that if we measure energy in units of and distance in units of , then the Schrödinger equation becomes:
,
and the energy eigenfunctions and eigenvalues become
where are the
Hermite polynomialsIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, where they give rise to the eigenstates of the quantum...
.
To avoid confusion, we will not adopt these natural units in this article. However, they frequently come in handy when performing calculations.
Characteristic impedance
The standard definition of the wave impedance of harmonic oscillator is:
where is the natural scale of oscillator length. In the general case the wave characteristic impedance will be:
This value is similar to the von Klitzing resistance constant
Note that, the wave impedance for Planck scale will be:
where is the
Planck massIn physics, the Planck mass is the unit of mass in the system of natural units known as Planck units. It is defined so that...
.
For electron mass the relationship will be:
Diatomic molecules
In diatomic molecules, the natural frequency is given by
where
ω = 2π
f is the angular frequency,
k is the force constant of the bond, and
μ is the
reduced massReduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...
.
N-dimensional harmonic oscillator
The one-dimensional harmonic oscillator is readily generalizable to
N dimensions, where
N = 1, 2, 3, ... . In one dimension, the position of the particle was specified by a single
coordinateIn mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex...
,
x. In
N dimensions, this is replaced by
N position coordinates, which we label
x1, ...,
xN. Corresponding to each position coordinate is a momentum; we label these
p1, ...,
pN. The canonical commutation relations between these operators are
.
The Hamiltonian for this system is
.
As the form of this Hamiltonian makes clear, the
N-dimensional harmonic oscillator is exactly analogous to
N independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities
x1, ...,
xN would refer to the positions of each of the
N particles. This is a happy property of the
r2 potential, which allows the potential energy to be separated into terms depending on one coordinate each.
This observation makes the solution straightforward. For a particular set of quantum numbers {
n} the energy eigenfunctions for the
N-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:
In the ladder operator method, we define
N sets of ladder operators,
.
By a procedure analogous to the one-dimensional case, we can then show that each of the
ai and
a†i operators lower and raise the energy by ℏω respectively. The Hamiltonian is
This Hamiltonian is invariant under the dynamic symmetry group
U(N) (the unitary group in
N dimensions), defined by
where is an element in the defining matrix representation of
U(N).
The energy levels of the system are
.
As in the one-dimensional case, the energy is quantized. The ground state energy is
N times the one-dimensional energy, as we would expect using the analogy to
N independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In
N-dimensions, except for the ground state, the energy levels are
degenerate, meaning there are several states with the same energy.
The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Define
n =
n1 +
n2 +
n3. All states with the same
n will have the same energy. For a given
n, we choose a particular
n1. Then
n2 +
n3 =
n −
n1. There are
n −
n1 + 1 possible groups {
n2,
n3}.
n2 can take on the values 0 to
n −
n1, and for each
n2 the value of
n3 is fixed. The degree of degeneracy therefore is:
Formula for general
N and
n [
gn being the dimension of the symmetric irreducible
nth power representation of the unitary group
U(N)]:
The special case
N = 3, given above, follows directly from this general equation.
Example: 3D isotropic harmonic oscillator
The Schrödinger equation of a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables, see this article for the present case. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with the
spherically symmetric potentialIn quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a potential which has spherical symmetry...
where is the mass of the problem. Because
m will be used below for the magnetic quantum number, mass is indicated by , instead of
m, as earlier in this article.
The solution reads
where is a normalization constant. are generalized Laguerre polynomials. The order
k of the polynomial is a non-negative integer. is a
spherical harmonic functionIn mathematics, the spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre Simon...
. is the reduced
Planck constantThe Planck constant , also called Planck's constant, is a physical constant used to describe the sizes of quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory...
: .
The energy eigenvalue is
The energy is usually described by the single
quantum numberQuantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
Because
k is a non-negative integer, for every even
n we have and for every odd
n we have . The magnetic quantum number
m is an integer satisfying , so for every
n and
l there are
2l+1 different
quantum stateIn quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be...
s, labeled by
m. Thus, the degeneracy at level
n is
where the sum starts from 0 or 1, according to whether
n is even or odd.
This result is in accordance with the dimension formula above.
Coupled harmonic oscillators
In this problem, we consider
N equal masses which are connected to their neighbors by springs, in the limit of large
N. The masses form a linear chain in one dimension, or a regular lattice in two or three dimensions.
As in the previous section, we denote the positions of the masses by
x1,
x2, ..., as measured from their equilibrium positions (i.e.
xk = 0 if particle
k is at its equilibrium position.) In two or more dimensions, the
xs are
vectorA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
quantities. The Hamiltonian of the total system is
The potential energy is summed over "nearest-neighbor" pairs, so there is one term for each spring.
Remarkably, there exists a coordinate transformation to turn this problem into a set of independent harmonic oscillators, each of which corresponds to a particular collective distortion of the lattice. These distortions display some particle-like properties, and are called
phononIn physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's...
s. Phonons occur in the ionic lattices of many
solidMatter is generally found in three different forms: solid, liquid, and gas . The solid state of matter is characterized by a distinct structural rigidity and resistance to deformation . Most solids have high values both of Young's modulus and of the shear modulus of elasticity...
s, and are extremely important for understanding many of the phenomena studied in solid state physics.
See also
- Gas in a harmonic trap
The results of the quantum harmonic oscillator can be used to look at the equilibrium situation for a quantum ideal gas in a harmonic trap, which is a harmonic potential containing a large number particles that do not interact with each other except for instantaneous thermalizing collisions...
- Creation and annihilation operators
Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one...
- Coherent state
In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system...
- Morse potential
The Morse potential, named after physicist Philip M. Morse, is a convenient model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond...
- Hooke's atom
Hooke's atom, also known as harmonium, refers to an artificial helium-like atom where the Coulombic electron-nucleus interaction potential isreplaced by a harmonic potential...
- Bertrand's theorem
In classical mechanics, Bertrand's theorem states that only two types of potentials produce stable, closed orbits: an inverse-square central force such as the gravitational or electrostatic potentialand the radial harmonic oscillator potential...
External links