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The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known. [1] [2] [3]
The Hamiltonian of the particle is: where m is the particle's mass, k is the force constant, is the angular frequency of the oscillator, is the position operator (given by x in the coordinate basis), and is the momentum operator (given by in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law. [4]
The time-independent Schrödinger equation is, where denotes a real number (which needs to be determined) that will specify a time-independent energy level, or eigenvalue, and the solution denotes that level's energy eigenstate. [5]
Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function , using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions, [6] [7]
The functions Hn are the physicists' Hermite polynomials,
The corresponding energy levels are [8] The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be and owing to the symmetry of the problem, whereas:
The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of which is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction. [9]
This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of ħω) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the n = 0 state, called the ground state) is not equal to the minimum of the potential well, but ħω/2 above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle.
The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.
The "ladder operator" method, developed by Paul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation. [10] It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjoint a†, Note these operators classically are exactly the generators of normalized rotation in the phase space of and , i.e they describe the forwards and backwards evolution in time of a classical harmonic oscillator.[ clarification needed ]
These operators lead to the following representation of and ,
The operator a is not Hermitian, since itself and its adjoint a† are not equal. The energy eigenstates |n⟩, when operated on by these ladder operators, give
From the relations above, we can also define a number operator N, which has the following property:
The following commutators can be easily obtained by substituting the canonical commutation relation,
and the Hamilton operator can be expressed as
so the eigenstates of N are also the eigenstates of energy. To see that, we can apply to a number state :
Using the property of the number operator :
we get:
Thus, since solves the TISE for the Hamiltonian operator , is also one of its eigenstates with the corresponding eigenvalue:
QED.
The commutation property yields
and similarly,
This means that a acts on |n⟩ to produce, up to a multiplicative constant, |n–1⟩, and a† acts on |n⟩ to produce |n+1⟩. For this reason, a is called an annihilation operator ("lowering operator"), and a† a creation operator ("raising operator"). The two operators together are called ladder operators.
Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with ħω less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = −∞. However, since
the smallest eigenvalue of the number operator is 0, and
In this case, subsequent applications of the lowering operator will just produce zero, instead of additional energy eigenstates. Furthermore, we have shown above that
Finally, by acting on |0⟩ 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 given in the preceding section.
Arbitrary eigenstates can be expressed in terms of |0⟩, [11]
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation . In the position representation, this is the first-order differential equation whose solution is easily found to be the Gaussian [nb 1] Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the Hermite functions energy eigenstates constructed by the ladder method form a complete orthonormal set of functions. [12]
Explicitly connecting with the previous section, the ground state |0⟩ in the position representation is determined by , hence so that , and so on.
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 energy is measured in units of ħω and distance in units of √ħ/(mω), then the Hamiltonian simplifies to while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half, where Hn(x) are the Hermite polynomials.
To avoid confusion, these "natural units" will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.
For example, the fundamental solution (propagator) of H − i∂t, the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel, [13] [14] where K(x,y;0) = δ(x − y). The most general solution for a given initial configuration ψ(x,0) then is simply
The coherent states (also known as Glauber states) of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty σxσp = ℏ⁄2, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality. [15]
The coherent states are indexed by and expressed in the |n⟩ basis as
Since coherent states are not energy eigenstates, their time evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter α instead: .
Because and via the Kermack-McCrae identity, the last form is equivalent to a unitary displacement operator acting on the ground state: . Calculating the expectation values:
where is the phase contributed by complex α. These equations confirm the oscillating behavior of the particle.
The uncertainties calculated using the numeric method are:
which gives . Since the only wavefunction that can have lowest position-momentum uncertainty, , is a gaussian wavefunction, and since the coherent state wavefunction has minimum position-momentum uncertainty, we note that the general gaussian wavefunction in quantum mechanics has the form:Substituting the expectation values as a function of time, gives the required time varying wavefunction.
The probability of each energy eigenstates can be calculated to find the energy distribution of the wavefunction:
which corresponds to Poisson distribution.
When n is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy En can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through asymptotics of the Hermite polynomials, and also through the WKB approximation.
The frequency of oscillation at x is proportional to the momentum p(x) of a classical particle of energy En and position x. Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to p(x), reflecting the length of time the classical particle spends near x. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region, to be approximately This is also given, asymptotically, by the integral
In the phase space formulation of quantum mechanics, eigenstates of the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution.
The Wigner quasiprobability distribution for the energy eigenstate |n⟩ is, in the natural units described above,[ citation needed ] where Ln are the Laguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner map.
Meanwhile, the Husimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have This claim can be verified using the Segal–Bargmann transform. Specifically, since the raising operator in the Segal–Bargmann representation is simply multiplication by and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply . At this point, we can appeal to the formula for the Husimi Q function in terms of the Segal–Bargmann transform.
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 coordinate, 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 convenient 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 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 an analogous procedure 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 ground 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 pairs {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 n-th power representation of the unitary group U(N)]: The special case N = 3, given above, follows directly from this general equation. This is however, only true for distinguishable particles, or one particle in N dimensions (as dimensions are distinguishable). For the case of N bosons in a one-dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer n using integers less than or equal to N.
This arises due to the constraint of putting N quanta into a state ket where and , which are the same constraints as in integer partition.
The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with a different spherically symmetric potential where μ is the mass of the particle. Because m will be used below for the magnetic quantum number, mass is indicated by μ, instead of m, as earlier in this article.
The solution to the equation is: [16] where
are generalized Laguerre polynomials; The order k of the polynomial is a non-negative integer;
The energy eigenvalue is The energy is usually described by the single quantum number
Because k is a non-negative integer, for every even n we have ℓ = 0, 2, …, n − 2, n and for every odd n we have ℓ = 1, 3, …, n − 2, n . The magnetic quantum number m is an integer satisfying −ℓ ≤ m ≤ ℓ, so for every n and ℓ there are 2ℓ + 1 different quantum states, 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, and amounts to the dimensionality of a symmetric representation of SU(3), [17] the relevant degeneracy group.
The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and 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. xi = 0 if the particle i is at its equilibrium position). In two or more dimensions, the xi are vector quantities. The Hamiltonian for this system is
where m is the (assumed uniform) mass of each atom, and xi and pi are the position and momentum operators for the i th atom and the sum is made over the nearest neighbors (nn). However, it is customary to rewrite the Hamiltonian in terms of the normal modes of the wavevector rather than in terms of the particle coordinates so that one can work in the more convenient Fourier space.
We introduce, then, a set of N "normal coordinates" Qk, defined as the discrete Fourier transforms of the xs, and N "conjugate momenta" Π defined as the Fourier transforms of the ps,
The quantity kn will turn out to be the wave number of the phonon, i.e. 2π divided by the wavelength. It takes on quantized values, because the number of atoms is finite.
This preserves the desired commutation relations in either real space or wave vector space
From the general result it is easy to show, through elementary trigonometry, that the potential energy term is where
The Hamiltonian may be written in wave vector space as
Note that the couplings between the position variables have been transformed away; if the Qs and Πs were hermitian (which they are not), the transformed Hamiltonian would describe Nuncoupled harmonic oscillators.
The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the (N + 1)-th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
The upper bound to n comes from the minimum wavelength, which is twice the lattice spacing a, as discussed above.
The harmonic oscillator eigenvalues or energy levels for the mode ωk are
If we ignore the zero-point energy then the levels are evenly spaced at
So an exact amount of energy ħω, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In analogy to the photon case when the electromagnetic field is quantised, the quantum of vibrational energy is called a phonon.
All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described elsewhere. [18]
In the continuum limit, a → 0, N → ∞, while Na is held fixed. The canonical coordinates Qk devolve to the decoupled momentum modes of a scalar field, , whilst the location index i (not the displacement dynamical variable) becomes the parameter x argument of the scalar field, .
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
The Schrödinger equation is a partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle in physics, a phonon is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.
In physics, the Heisenberg picture or Heisenberg representation is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.
Creation operators 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. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization. They were introduced by Paul Dirac.
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position and momentum of a particle, and the (dimension-less) electric field in the amplitude and in the mode of a light wave. The product of the standard deviations of two such operators obeys the uncertainty principle:
In physics, the Lamb shift, named after Willis Lamb, is an anomalous difference in energy between two electron orbitals in a hydrogen atom. The difference was not predicted by theory and it cannot be derived from the Dirac equation, which predicts identical energies. Hence the Lamb shift is a deviation from theory seen in the differing energies contained by the 2S1/2 and 2P1/2 orbitals of the hydrogen atom.
In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.
Quantum noise is noise arising from the indeterminate state of matter in accordance with fundamental principles of quantum mechanics, specifically the uncertainty principle and via zero-point energy fluctuations. Quantum noise is due to the apparently discrete nature of the small quantum constituents such as electrons, as well as the discrete nature of quantum effects, such as photocurrents.
The Jaynes–Cummings model is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.
In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, thus known as Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom interacts with a single-mode field confined to a limited volume V in an optical cavity. Spontaneous emission is a consequence of coupling between the atom and the vacuum fluctuations of the cavity field.
In 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. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:
The quantization of the electromagnetic field is a procedure in physics turning Maxwell's classical electromagnetic waves into particles called photons. Photons are massless particles of definite energy, definite momentum, and definite spin.
In ion trapping and atomic physics experiments, the Lamb Dicke regime is a quantum regime in which the coupling between an ion or atom's internal qubit states and its motional states is sufficiently small so that transitions that change the motional quantum number by more than one are strongly suppressed.
In quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters, between a state containing few electromagnetic excitations and a superradiant state with many electromagnetic excitations trapped inside the emitters. The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters.
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.