Definitions

Old quantum theory

The old quantum theory was a collection of results from the years 1900-1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics.

The old-quantum theory lives on as an approximation technique in quantum mechanics, called the WKB method. Semi-classical approximations were a popular research subject in the 1970s and 1980s, after Gutzwiller discovered a semi-classical description for systems which are classically chaotic (see quantum chaos).

Basic principles

The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition:


int p_i dq_i = n_i h

where the $P_i$ are the momentum of the system and the $q_i$ are the coordinates. The quantum numbers $n_i$ are integers and the integral is taken over one period of the motion. The integral is an area in phase space, which is a quantity called the action, which is quantized in units of Planck's constant. For this reason, Planck's constant was often called the quantum of action.

In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates $q_i$ in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.

The motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.

Examples

Harmonic oscillator

The simplest system in the old quantum theory is the Harmonic oscillator, whose Hamiltonian is:


H= {p^2 over 2m} + {momega^2 q^2over 2}

The level sets of H are the orbits, and the quantum condition is that the area enclosed by an orbit in phase space is an integer. It follows that the energy is quantized according to the Planck rule:


E= nhbar omega ,

a result which was known well before, and used to formulate the old quantum condition.

The thermal properties of a quantized oscillator may be found by averaging the energy in each of the discrete states assuming that they are occupied with a Boltzmann weight:


U = {sum_n hbaromega n e^{-beta nhbaromega} over sum_n e^{-beta n hbaromega}} = {hbar omega e^{-betahbaromega} over 1 - e^{-betahbaromega}}

When $beta$, the reciprocal of the temperature, is very large so that the system is very cold, the amount of energy in an oscillator approaches zero exponentially fast. Thermal fluctuations forbid a transition in the oscillator of even one quantum. This means that the specific heat of a cold system of vibrating quantum atoms approaches zero at low temperatures. In classical mechanics the amount of energy in an oscillator is linearly proportional to the temperature, so that the specific heat is constant.

Experimentally, solids have a low specific heat at low temperatures, approaching zero at absolute zero. That this is true for all material systems is an observation codified in the third law of thermodynamics. The contradiction between classical mechanics and the specific heat of cold materials was noted by James Clerk Maxwell in the 19th century, and remained a deep puzzle for those who advocated an atomic theory of matter. To resolve it, Einstein in 1906 proposed that atomic motion is quantized, the first application of quantum theory to a mechanical system. A short while later, Debye gave a quantitative theory of solid specific heats in terms of quantized oscillators with various frequencies (see Einstein solid and Debye model).

One dimensional potential

One dimensional problems are easy to solve. At any energy E, the value of the momentum p is found from the conservation equation:


sqrt{2m(E - V(q))} = p

which is integrated over all values of q between the classical turning points, the places where the momentum vanishes. The integral is easiest for a particle in a box of length L, where the quantum condition is:


2int_0^L p = nh

which gives the allowed momenta:


p= {nh over 2L}

and the energy levels


E= {n^2 h^2 over 8mL^2}

Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force F binding a particle to an impenetrable wall. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.


2 int sqrt{2mE - Fx} = n h

This integral is the area under a (sideways) parabola, which is 4/3 the area of an inscribed triangle of height $sqrt\left\{2mE\right\}$ and base $2mE/F$. So that the quantum condition is:


{2sqrt{2}over 3} { m^{3/2}over sqrt{F} } E^{3/2} = n h Which determines the energy levels.

Rotator

Another simple system is the rotator. A rotator consists of a mass M at the end of a massless rigid rod of length R and in two dimensions has the Lagrangian:


L = {mR^2 over 2} dottheta^2

which determines that the momentum J conjugate to $theta$, the polar angle, $J = mR^2 dottheta$. The old quantum condition requires that J multiplied by the period of $theta$ is an integer multiple of Planck's constant:


2pi J = n h ,

the angular mometum to be an integer multiple of $scriptstyle hbar$. In the Bohr model, this restriction imposed on circular orbits was enough to determine the energy levels.

In three dimensions, a rigid rotator can be described by two angles--- $theta$ and $phi$, where $theta$ is the inclination relative to an arbitrarily chosen z-axis while $phi$ is the rotator angle in the projection to the x-y plane. The kinetic energy is again the only contribution to the Lagrangian:


L = {1over 2} dottheta^2 + {1over 2} (sin(theta)dotphi)^2 ,

And the conjugate momenta are $scriptstyle P_theta = dottheta$ and $p_phi=sin\left(theta\right)^2 dotphi$. The equation of motion for $phi$ is trivial: $p_phi$ is a constant value l_phi.


p_phi = l_phi ,

which is the z-component of the angular momentum. The quantum condition demands that the integral of the constant $l_phi$ as $phi$ varies from 0 to $2pi$ is an integer multiple of h:


l_phi = m hbar ,

And m is called the magnetic quantum number, because the z component of the angular momentum is the magnetic moment of the rotator along the z direction in the case where the particle at the end of the rotator is charged.

Since the three dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator. The two quantum conditions restrict the total angular momentum and the z-component of the angular momentum to be the integers l,m. This condition is reproduced in modern quantum mechanics, but in the era of the old quantum theory it led to a paradox. How can the orientation of the angular momentum relative to the arbitrarily chosen z-axis be quantized? This mystery was resolved by quantum superposition.

Hydrogen atom

The angular part of the Hydrogen atom is just the rotor, and gives the quantum numbers l and m. The only remaining variable is the radial coordinate, which executes a periodic one dimensional potential motion, which can be solved.

For a fixed value of the total angular momentum L, the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants):


H= { p^2 over 2 } + {l^2 over 2 r^2 } - {1over r}

Fixing the energy to be constant and solving for the radial momentum p, the quantum condition integral is:


2 int sqrt{2E - {l^2over r^2} - { 1over r}} = k h

which is elementary, and gives a new quantum number k which determines the energy in combination with l. The energy is:


E= -{1 over (k + l)^2}

and it only depends on the sum of k and l, which is the principal quantum number n. Since k is positive, the allowed values of l for any given n are no bigger than n. The energies reproduce those in the Bohr model, except with the correct quantum mechanical multiplicities, with some ambiguity at the extreme values.

The semiclassical hydrogen atom is called the Sommerfeld model, and its orbits are ellipses of various sizes at discrete inclinations. The sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values, a result which seems to contradict rotational invariance but which was confirmed by the Stern-Gerlach experiment.

DeBroglie waves

In 1905, Einstein noted that the entropy of the quantized electromagnetic field oscillators in a box is, for short wavelength, equal to the entropy of a gas of point particles in the same box. The number of point particles is equal to the number of quanta. Einstein concluded that the quanta were localizable objects, particles of light, and named them photons.

Einstein's theoretical argument was based on thermodynamics, on counting the number of states, and so was not completely convincing. Nevertheless, he concluded that light had attributes of both waves and particles, more precisely that an electromagnetic standing wave with frequency $omega$ with the quantized energy:


E = nhbaromega , should be thought of as consisting of n photons each with an energy $scriptstyle hbaromega$. Einstein could not describe how the photons were related to the wave.

The photons have momentum as well as energy, and the momentum had to be $scriptstyle hbar k$ where $k$ is the wavenumber of the electromagnetic wave. This is required by relativity, because the momentum and energy form a four-vector, as do the frequency and wave-number.

In 1924, as a PhD candidate, Louis DeBroglie proposed a new interpretation of the quantum condition. He suggested that all matter, electrons as well as photons, are described by waves obeying the relations.


p = hbar k or, expressed in terms of wavelength $lambda$ instead,

p = {h over lambda} He then noted that the quantum condition:

int p dx = hbar int k dx = 2pihbar n counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple of $2pi$. Expressed in wavelengths, the number of wavelengths along a classical orbit must be an integer. This is the condition for constructive interference, and it explained the reason for quantized orbits--- the matter waves make standing waves only at discrete frequencies, at discrete energies.

For example, for a particle confined in a box, a standing wave must fit an integer number of wavelengths between twice the distance between the walls. The condition becomes:


nlambda = 2L , so that the quantized momenta are:

p = {hbar 4pi L over n} reproducing the old quantum energy levels.

This development was given a more mathematical form by Einstein, who noted that the phase function for the waves: $theta\left(J,x\right)$ in a mechanical system should be identified with the solution to the Hamilton-Jacobi equation, an equation which even Hamilton considered to be a short-wavelength limit of a wave mechanics.

These ideas led to the development of Schrödinger equation.

Kramers transition matrix

The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic. It did not deal with the emission and absorption of radiation. Nevertheless, Hendrik Kramers was able to find heuristics for describing how emission and absorption should be calculated.

Kramers suggested that the orbits of a quantum system should be Fourier analyzed, decomposed into harmonics at multiples of the orbit frequency:


X_n(t) = sum_{k=-infty}^{infty} e^{ikomega t} X_{n;k}

The index n describes the quantum numbers of the orbit, it would be n-l-m in the Sommerfeld model. The frequency $omega$ is the angular frequency of the orbit $scriptstyle 2pi/T_n$ while k is an index for the fourier mode. Bohr had suggested that the k-th harmonic of the classical motion correspond to the transition from level n to level n-k.

Kramers proposed that the transition between states were analogous to classical emission of radiation, which happens at frequencies at multiples of the orbit frequencies. The rate of emission of radiation is proportional to $|X_k|^2$, as it would be in classical mechanics. The description was approximate, since the Fourier components did not have frequencies that exactly match the energy spacings between levels.

This idea led to the development of matrix mechanics.

History

The old quantum theory was sparked by the work of Max Planck on the emission and absorption of light, and began in earnest after the work of Albert Einstein on the specific heats of solids. Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly.

In 1913, Neils Bohr identified the correspondence principle and used it to formulate a model of the Hydrogen atom which explained the line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Lorentz and Einstein. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.

Throughout the 1910s and well into the 1920s, many problems were attacked using the old quantum theory with mixed results. Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers. Max Planck introduced the zero point energy and Arnold Sommerfeld semiclassically quantized the relativistic hydrogen atom. Hendrik Kramers explained the stark effect. Bose and Einstein gave the correct quantum statistics for photons.

Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which were extended in collaboration with Werner Heisenberg to a semiclassical matrix-like description of atomic transition probabilities. Heisenberg went on to reformulate all of quantum theory in terms of a version of these transition matrices, creating Matrix mechanics.

In 1924, Louis DeBroglie introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Albert Einstein a short time later. In 1926 Erwin Schrödinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. Schrödinger's wave mechanics developed separately from matrix mechanics until Schrödinger and others proved that the two methods predicted the same experimental consequences. Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called transformation theory. Matrix mechanics and wave mechanics put an end to the era of the old-quantum theory.

References

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