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The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because, as in classical mechanics, a wide variety of physical situations can be reduced to it either exactly or approximately. In particular, a system near an equilibrium configuration can often be described in terms of one or more harmonic oscillators. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution is known. Jump to: navigation, search Fig. ... Jump to: navigation, search A harmonic oscillator is either a mechanical system in which there exists a returning force F directly proportional to the displacement x from a given equilibrium position, i. ...


The following discussion of the quantum harmonic oscillator relies on the article mathematical formulation of quantum mechanics. One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces. ...

Contents


One-dimensional harmonic oscillator

Hamiltonian and energy eigenstates

Wavefunctions ψn(x) for the first six bound eigenstates, n = 0 to 5. The horizontal axis shows the position x. The graphs are not normalised.
Enlarge
Wavefunctions ψn(x) for the first six bound eigenstates, n = 0 to 5. The horizontal axis shows the position x. The graphs are not normalised.
Probability densities |ψn(x)|² for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities.
Probability densities n(x)|² for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities.

In the one-dimensional harmonic oscillator problem, a particle of mass m is subject to a potential V(x) = (1/2)mω2 x2. The Hamiltonian of the particle is: Image File history File links Download high resolution version (430x1378, 12 KB)Solutions to the schrodinger equation under a harmonic potential Created with a C program of my own creation. ... Image File history File links Download high resolution version (430x1378, 12 KB)Solutions to the schrodinger equation under a harmonic potential Created with a C program of my own creation. ... Image File history File links Picture of wavefunctions (and energy levels) of quantum harmonic oscillator, using color scale for probability density. ... Image File history File links Picture of wavefunctions (and energy levels) of quantum harmonic oscillator, using color scale for probability density. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

H = {p^2 over 2m} + {1over 2} m omega^2 x^2

where x is the position operator, and p is the momentum operator (p = -i hbar partial / partial x). The first term represents the kinetic energy of the particle, and the second term represents the potential energy in which it resides. In order to find the energy levels and the corresponding energy eigenstates, we must solve the time-independent Schrödinger equation, Jump to: navigation, search Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. ... In physics, momentum is a physical quantity related to the velocity and mass of an object. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

H left| psi rightrangle = E left| psi rightrangle.

We can solve the differential equation in the coordinate basis, using a power series method. It turns out that there is a family of solutions, In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

leftlangle x | psi_n rightrangle = frac{1}{sqrt{2^n n!}} left(frac{momega}{pi hbar}right)^{1/4} hbox{exp} left(- frac{momega x^2}{2 hbar} right) H_nleft(sqrt{frac{momega}{hbar}} x right)
n = 0, 1, 2, ldots

The first six solutions (n = 0 to 5) are shown on the right. The functions Hn(θ) are the Hermite polynomials: In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ...

H_n(x)=(-1)^n e^{x^2}frac{d^n}{dx^n}e^{-x^2}

They should not be confused with the Hamiltonian, which is unfortunately also denoted by H. The corresponding energy levels are

E_n = hbar omega left(n + {1over 2}right).

This energy spectrum is noteworthy for two reasons. Firstly, the energies are "quantized", and may only take the discrete values of hbaromega times 1/2, 3/2, 5/2, and so forth. 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 hbaromega/2, which is called the "ground state energy" or zero-point energy. 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 gravity. Unsolved problems in physics: Is zero-point energy physical, and if so, are there any practical applications and does it have any connection with dark energy? In a quantum mechanical system such as the particle in a box or the quantum harmonic oscillator, the lowest possible energy is called the... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ...


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 principle is thus satisfied. In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...


Ladder operator method

The power series solution, though straightforward, is rather tedious. The "ladder operator" method, due to Paul Dirac, 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 theory. Following this approach, we define the operators a and its adjoint a Jump to: navigation, search Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Jump to: navigation, search Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In mathematics, the term adjoint applies in several situations. ...

begin{matrix} a &=& sqrt{momega over 2hbar} left(x + {i over m omega} p right)  a^{dagger} &=& sqrt{m omega over 2hbar} left( x - {i over m omega} p right) end{matrix}

The operator a is not Hermitian since it and its adjoint a are not equal. On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...


In deriving the form of a, we have used the fact that the operators x and p, which represent observables, are Hermitian.


The x and p operators obey the following identity, known as the canonical commutation relation: In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...

left[x , p right] = ihbar.

The square brackets in this equation are a commonly-used notational device, known as the commutator, defined as For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how badly a certain binary operation fails to be commutative. ...

left[A , B right] equiv AB - BA.

Using the above, we can prove the identities

H = hbar omega left(a^{dagger}a + 1/2right)
left[a , a^{dagger} right] = 1.

Now, let left|psi_Erightrangle denote an energy eigenstate with energy E. The inner product of any ket with itself must be non-negative, so

left(a left|psi_E rightrangle, a left|psi_E rightrangleright) = leftlanglepsi_E right| a^dagger a left| psi_E rightrangle ge 0.

Expressing aa in terms of the Hamiltonian:

leftlanglepsi_E right| {H over hbar omega} - {1 over 2} left|psi_Erightrangle = left({E over hbar omega} - {1 over 2} right) ge 0,

so that E ge hbar omega / 2. Note that when (a left| psi_E right rangle) is the zero ket (i.e. a ket with length zero), the inequality is saturated, so that E = hbar omega / 2. 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:

begin{matrix} left[H , a right] &=& - hbar omega a  left[H , a ^daggerright] &=& hbar omega a^dagger end{matrix}.

Thus, provided (a left| psi_E right rangle) is not the zero ket,

begin{matrix} H (a left| psi_E rightrangle) &=& (left[H,aright] + a H) left|psi_Erightrangle  &=& (- hbaromega a + a E) left|psi_Erightrangle  &=& (E - hbaromega) (aleft|psi_Erightrangle) end{matrix}.

Similarly, we can show that

H (a^dagger left| psi_E rightrangle) = (E + hbaromega) (a^dagger left| psi_E rightrangle).

In other words, a acts on an eigenstate of energy E to produce, up to a multiplicative constant, another eigenstate of energy E - hbar omega, and a acts on an eigenstate of energy E to produce an eigenstate of energy E + hbar omega. For this reason, a is called a "lowering operator", and a a "raising operator". The two operators together are called "ladder operators". 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 hbar omega, 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 E ge hbar omega / 2. Therefore, there must be a ground-state energy eigenstate, which we label left| 0 right rangle (not to be confused with the zero ket), such that

a left| 0 rightrangle = 0 hbox{(zero ket)}.

In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstate. Furthermore, we have shown above that

H left|0rightrangle = (hbaromega/2) left|0rightrangle

Finally, by acting on left| 0 right rangle with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates left{left| 0 right rangle, left| 1 right rangle, left| 2 right rangle, ... , left| n right rangle, ...right}, such that

H left|nrightrangle = hbaromega (n + 1/2) left|nrightrangle

which matches the energy spectrum which we gave in the preceding section.


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 hbar omega and distance in units of left(hbar / left(m omegaright)right)^{1/2}, then the Schrödinger equation becomes: Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ...

H = - {1over2} {partial^2 over partial u^2 } + {1 over 2} u^2,

and the energy eigenfunctions and eigenvalues become

leftlangle x | psi_n rightrangle = {1 over sqrt{2^n n!}} pi^{-1/4} hbox{exp} (-u^2 / 2) H_n(u)
E_n = n + {1over 2}.

To avoid confusion, we will not adopt these natural units in this article. However, they frequently come in handy when performing calculations.


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 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 See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

begin{matrix} left[x_i , p_j right] &=& ihbardelta_{i,j}  left[x_i , x_j right] &=& 0  left[p_i , p_j right] &=& 0 end{matrix}.

The Hamiltonian for this system is

H = sum_{i=1}^N left( {p_i^2 over 2m} + {1over 2} m omega^2 x_i^2 right).

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:

langle mathbf{x}|psi_{{n}}rangle =prod_{i=1}^Nlangle x_i|psi_{n_i}rangle

In the ladder operator method, we define N sets of ladder operators,

begin{matrix} a_i &=& sqrt{momega over 2hbar} left(x_i + {i over m omega} p_i right)  a^{dagger}_i &=& sqrt{m omega over 2hbar} left( x_i - {i over m omega} p_i right) end{matrix}.

By a procedure analogous to the one-dimensional case, we can then show that each of the ai and ai operators lower and raise the energy by ℏω respectively. The energy levels of the system are

E = hbar omega left[(n_1 + cdots + n_N) + {Nover 2}right].
, n_i = 0, 1, 2, dots

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 {n2n3}. 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:

g_n = sum_{n_1=0}^n n - n_1 + 1 = sum_{n_1=0}^n n + 1 - sum_{n_1=0}^n n_1 = (n+1)(n+1) - frac{n(n+1)}{2} = frac{(n+1)(n+2)}{2}

Related problems

The quantum harmonic oscillator can be extended in many interesting ways. We will briefly discuss two of the more important extensions, the anharmonic oscillator and coupled harmonic oscillators.


Anharmonic oscillator

As mentioned in the introduction, a system residing "near" the minimum of some potential may be treated as a harmonic oscillator. In this approximation, we Taylor-expand the potential energy around the minimum and discard terms of third or higher order, resulting in an approximate quadratic potential. Once we have studied the system in this approximation, we may wish to investigate the corrections due to the discarded higher-order terms, particularly the third-order term. Jump to: navigation, search As the degree of the Taylor series rises, it approaches the correct function. ...


The anharmonic oscillator Hamiltonian is the harmonic oscillator Hamiltonian with an additional x3 potential:

H = {p^2 over 2m} + {1over 2} m omega^2 x^2 + lambda x^3

If the harmonic approximation is valid, the coefficient λ is small compared to the quadratic term. We may therefore use perturbation theory to determine the corrections to the states and energy levels imposed by the anharmonic term. This task may be simplified by using the ladder operators to rewrite the anharmonic term as In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ...

lambda left({hbar over 2momega}right)^{3over2} (a + a^dagger)^3.

It turns out that the correction to the energies vanish to first-order in λ. The second-order corrections are given by the usual formula in perturbation theory:

Delta E^{(2)} = lambda^2 leftlangle psi_E right| x^3 {1 over E - H_0} x^3 left| psi_E rightrangle.

This is straightforward, though tedious, to evaluate.


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 vector quantities. The Hamiltonian of the total system is Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

H = sum_{i=1}^N {p_i^2 over 2m} + {1over 2} m omega^2 sum_{{ij} (nn)} (x_i - x_j)^2

The potential energy is summed over "nearest-neighbor" pairs, so there is one term for each spring.


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 phonons. Phonons occur in the ionic lattices of many solids, and are extremely important for understanding many of the phenomena studied in solid state physics. A phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... In jewelry, a solid gold piece is the alternative to gold-filled or gold-plated jewelry. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ...


See also

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 which do not interact with each other except for instantaneous thermalizing collisions. ... This article needs to be cleaned up to conform to a higher standard of quality. ... 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. ...

References

  • Griffiths, David J. (2004) Introduction to Quantum Mechanics (2nd ed.), Prentice Hall. ISBN 013805326X
  • Liboff, Richard L. (2002) Introductory Quantum Mechanics, Addison-Wesley. ISBN 0805387145

  Results from FactBites:
 
Quantum harmonic oscillator (1756 words)
The quantum harmonic oscillator is a quantum mechanical analogue of the classical harmonic oscillator.
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 quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem.
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