Last modified on 29 May 2009, at 23:01

Quantum Mechanics/Waves and Modes

Many misconceptions about quantum mechanics may be avoided if some concepts of field theory and quantum field theory like "normal mode" and "occupation" are introduced right from the start. They are needed for understanding the deepest and most interesting ideas of quantum mechanics anyway. Questions about this approach are welcome on the talk page.

Waves and modesEdit

A wave is a propagating disturbance in a continuous medium or a physical field. By adding waves or multiplying their amplitudes by a scale factor, superpositions of waves are formed. Waves must satisfy the superposition principle which states that they can go through each other without disturbing each other. It looks like there were two superimposed realities each carrying only one wave and not knowing of each other (that's what is assumed if one uses the superposition principle mathematically in the wave equations).

Examples are acoustic waves and electromagnetic waves (light), but also electronic orbitals, as explained below.

A standing wave is considered a one-dimensional concept by many students, because of the examples (waves on a spring or on a string) usually provided. In reality, a standing wave is a synchronous oscillation of all parts of an extended object at a definite frequency, in which the oscillation profile (in particular the nodes and the points of maximal oscillation amplitude) doesn't change. This is also called a normal mode of oscillation. The profile can be made visible in Chladni's figures and in vibrational holography. In unconfined systems, i.e. systems without reflecting walls or attractive potentials, traveling waves may also be chosen as normal modes of oscillation (see boundary conditions).

A phase shift of a normal mode of oscillation is a time shift scaled as an angle in terms of the oscillation period, e.g. phase shifts by 90° and 180° (or \pi/2 and \pi) are time shifts by the fourth and half of the oscillation period, respectively. This operation is introduced as another operation allowed in forming superpositions of waves (mathematically, it is covered by the phase factors of complex numbers scaling the waves).

  • Helmholtz ran an experiment which clearly showed the physical reality of resonances in a box. (He predicted and detected the eigenfrequencies.)
  • experiments with standing and propagating waves

Electromagnetic and electronic modesEdit

Max Planck, one of the fathers of quantum mechanics.

Planck was the first to suggest that the electromagnetic modes are not excited continuously but discretely by energy quanta h \nu proportional to the frequency. By this assumption, he could explain why the high-frequency modes remain unexcited in a thermal light source: The thermal exchange energy k_B T is just too small to provide an energy quantum h \nu if \nu is too large. Classical physics predicts that all modes of oscillation (2 degrees of freedom each) -- regardless of their frequency -- carry the average energy k_B T, which amounts to an infinite total energy (called ultraviolet catastrophe). This idea of energy quanta was the historical basis for the concept of occupations of modes, designated as light quanta by Einstein, also denoted as photons since the introduction of this term in 1926 by Gilbert N. Lewis.

An electron beam (accelerated in a cathode ray tube similar to TV) is diffracted in a crystal and diffraction patterns analogous to the diffraction of monochromatic light by a diffraction grating or of X-rays on crystals are observed on the screen. This observation proved de Broglie's idea that not only light, but also electrons propagate and get diffracted like waves. In the attracting potential of the nucleus, this wave is confined like the acoustic wave in a guitar corpus. That's why in both cases a standing wave (= a normal mode of oscillation) forms. An electron is an occupation of such a mode.

An optical cavity.

An electronic orbital is a normal mode of oscillation of the electronic quantum field, very similar to a light mode in an optical cavity being a normal mode of oscillation of the electromagnetic field. The electron is said to be an occupation of an orbital. This is the main new idea in quantum mechanics, and it is forced upon us by observations of the states of electrons in multielectron atoms. Certain fields like the electronic quantum field are observed to allow its normal modes of oscillation to be excited only once at a given time, they are called fermionic. If you have more occupations to place in this quantum field, you must choose other modes (the spin degree of freedom is included in the modes), as is the case in a carbon atom, for example. Usually, the lower-energy (= lower-frequency) modes are favoured. If they are already occupied, higher-energy modes must be chosen. In the case of light, the idea that a photon is an occupation of an electromagnetic mode was found much earlier by Planck and Einstein, see below.

Processes and particlesEdit

All processes in nature can be reduced to the isolated time evolution of modes and to (superpositions of) reshufflings of occupations, as described in the Feynman diagrams (since the isolated time evolution of decoupled modes is trivial, it is sometimes eliminated by a mathematical redefinition which in turn creates a time dependence in the reshuffling operations; this is called Dirac's interaction picture, in which all processes are reduced to (redefined) reshufflings of occupations). For example in an emission of a photon by an electron changing its state, the occupation of one electronic mode is moved to another electronic mode of lower frequency and an occupation of an electromagnetic mode (whose frequency is the difference between the frequencies of the mentioned electronic modes) is created.

Electrons and photons become very similar in quantum theory, but one main difference remains: electronic modes cannot be excited/occupied more than once (= Pauli exclusion principle) while photonic/electromagnetic modes can and even prefer to do so (= stimulated emission).

This property of electronic modes and photonic modes is called fermionic and bosonic, respectively. Two photons are indistinguishable and two electrons are also indistinguishable, because in both cases, they are only occupations of modes: all that matters is which modes are occupied. The order of the occupations is irrelevant except for the fact that in odd permutations of fermionic occupations, a negative sign is introduced in the amplitude.

Of course, there are other differences between electrons and photons:

  • The electron carries an electric charge and a rest mass while the photon doesn't.
  • In physical processes (see the Feynman diagrams), a single photon may be created while an electron may not be created without at the same time removing some other fermionic particle or creating some fermionic antiparticle. This is due to the conservation of charge.

Mode numbers, Observables and eigenmodesEdit

The system of modes to describe the waves can be chosen at will. Any arbitrary wave can be decomposed into contributions from each mode in the chosen system. For the mathematically inclined: The situation is analogous to a vector being decomposed into components in a chosen coordinate system. Decoupled modes or, as an approximation, weakly coupled modes are particlularly convenient if you want to describe the evolution of the system in time, because each mode evolves independently of the others and you can just add up the time evolutions. In many situations, it is sufficient to consider less complicated weakly coupled modes and describe the weak coupling as a perturbation.

In every system of modes, you must choose some (continuous or discrete) numbering (called "quantum numbers") for the modes in the system. In Chladni's figures, you can just count the number of nodal lines of the standing waves in the different space directions in order to get a numbering, as long as it is unique. For decoupled modes, the energy or, equivalently, the frequency might be a good idea, but usually you need further numbers to distinguish different modes having the same energy/frequency (this is the situation referred to as degenerate energy levels). Usually these additional numbers refer to the symmetry of the modes. Plane waves, for example -- they are decoupled in spatially homogeneous situations -- can be characterized by the fact that the only result of shifting (translating) them spatially is a phase shift in their oscillation. Obviously, the phase shifts corresponding to unit translations in the three space directions provide a good numbering for these modes. They are called the wavevector or, equivalently, the momentum of the mode. Spherical waves with an angular dependence according to the spherical harmonics functions (see the pictures) -- they are decoupled in spherically symmetric situations -- are similarly characterized by the fact that the only result of rotating them around the z-axis is a phase shift in their oscillation. Obviously, the phase shift corresponding to a rotation by a unit angle is part of a good numbering for these modes; it is called the magnetic quantum number m (it must be an integer, because a rotation by 360° mustn't have any effect) or, equivalently, the z-component of the orbital angular momentum. If you consider sharp wavepackets as a system of modes, the position of the wavepacket is a good numbering for the system. In crystallography, the modes are usually numbered by their transformation behaviour (called group representation) in symmetry operations of the crystal, see also symmetry group, crystal system.

The mode numbers thus often refer to physical quantities, called observables characterizing the modes. For each mode number, you can introduce a mathematical operation, called operator, that just multiplies a given mode by the mode number value of this mode. This is possible as long as you have chosen a mode system that actually uses and is characterized by the mode number of the operator. Such a system is called a system of eigenmodes, or eigenstates: Sharp wavepackets are no eigenmodes of the momentum operator, they are eigenmodes of the position operator. Spherical harmonics are eigenmodes of the magnetic quantum number, decoupled modes are eigenvalues of the energy operator etc. If you have a superposition of several modes, you just operate the operator on each contribution and add up the results. If you chose a different modes system that doesn't use the mode number corresponding to the operator, you just decompose the given modes into eigenmodes and again add up the results of the operator operating on the contributions. So if you have a superposition of several eigenmodes, say, a superposition of modes with different frequencies, then you have contributions of different values of the observable, in this case the energy. The superposition is then said to have an indefinite value for the observable, for example in the tone of a piano note, there is a superposition of the fundamental frequency and the higher harmonics being multiples of the fundamental frequency. The contributions in the superposition are usually not equally large, e.g. in the piano note the very high harmonics don't contribute much. Quantitatively, this is characterized by the amplitudes of the individual contributions. If there are only contributions of a single mode number value, the superposition is said to have a definite or sharp value.

  • the basics of wave-particle duality.
If you do a position measurement, the result is the occupation of a very sharp wavepacket being an eigenmode of the position operator. These sharp wavepackets look like pointlike objects, they are strongly coupled to each other, which means that they spread soon.

MeasurementsEdit

In measurements of such a mode number in a given situation, the result is an eigenmode of the mode number, the eigenmode being chosen at random from the contributions in the given superposition. All the other contributions are supposedly eradicated in the measurement -- this is called the wave function collapse and some features of this process are questionable and disputed. The probability of a certain eigenmode to be chosen is equal to the absolute square of the amplitude, this is called Born's probability law. This is the reason why the amplitudes of modes in a superposition are called "probability amplitudes" in quantum mechanics. The mode number value of the resulting eigenmode is the result of the measurement of the observable. Of course, if you have a sharp value for the observable before the measurement, nothing is changed by the measurement and the result is certain. This picture is called the Copenhagen interpretation. A different explanation of the measurement process is given by Everett's many-worlds theory; it doesn't involve any wave function collapse. Instead, a superposition of combinations of a mode of the measured system and a mode of the measuring apparatus (an entangled state) is formed, and the further time evolutions of these superposition components are independent of each other (this is called "many worlds").

As an example: a sharp wavepacket is an eigenmode of the position observable. Thus the result of measurements of the position of such a wavepacket is certain. On the other hand, if you decompose such a wavepacket into contributions of plane waves, i.e. eigenmodes of the wavevector or momentum observable, you get all kinds of contributions of modes with many different momenta, and the result of momentum measurements will be accordingly. Intuitively, this can be understood by taking a closer look at a sharp or very narrow wavepacket: Since there are only a few spatial oscillations in the wavepacket, only a very imprecise value for the wavevector can be read off (for the mathematically inclined reader: this is a common behaviour of Fourier transforms, the amplitudes of the superposition in the momentum mode system being the Fourier transform of the amplitudes of the superposition in the position mode system). So in such a state of definite position, the momentum is very indefinite. The same is true the other way round: The more definite the momentum is in your chosen superposition, the less sharp the position will be, and it is called Heisenberg's uncertainty relation.

Two different mode numbers (and the corresponding operators and observables) that both occur as characteristic features in the same mode system, e.g. the number of nodal lines in one of Chladni's figures in x direction and the number of nodal lines in y-direction or the different position components in a position eigenmode system, are said to commute or be compatible with each other (mathematically, this means that the order of the product of the two corresponding operators doesn't matter, they may be commuted). The position and the momentum are non-commuting mode numbers, because you cannot attribute a definite momentum to a position eigenmode, as stated above. So there is no mode system where both the position and the momentum (referring to the same space direction) are used as mode numbers.

The Schrödinger equation, the Dirac equation etc.Edit

As in the case of acoustics, where the direction of vibration, called polarization, the speed of sound and the wave impedance of the media, in which the sound propagates, are important for calculating the frequency and appearance of modes as seen in Chladni's figures, the same is true for electronic or photonic/electromagnetic modes: In order to calculate the modes (and their frequencies or time evolution) exposed to potentials that attract or repulse the waves or, equivalently, exposed to a change in refractive index and wave impedance, or exposed to magnetic fields, there are several equations depending on the polarization features of the modes:

  • Electronic modes (their polarization features are described by Spin 1/2) are calculated by the Dirac equation, or, to a very good approximation in cases where the theory of relativity is irrelevant, by the Schrödinger equation]] and the Pauli equation.
  • Photonic/electromagnetic modes (polarization: Spin 1) are calculated by Maxwell's equations (You see, 19th century already found the first quantum-mechanical equation! That's why it's so much easier to step from electromagnetic theory to quantum mechanics than from point mechanics).
  • Modes of Spin 0 would be calculated by the Klein-Gordon equation.

ConsequencesEdit

It is much easier and much more physical to imagine the electron in the atom to be not some tiny point jumping from place to place or orbiting around (there are no orbits, there are orbitals), but to imagine the electron being an occupation of an extended orbital and an orbital being a vibrating wave confined to the neighbourhood of the nucleus by its attracting force. That's why Chladni's figures of acoustics and the normal modes of electromagnetic waves in a resonator are such a good analogy for the orbital pictures in quantum physics. Quantum mechanics is a lot less weird if you see this analogy. The step from electromagnetic theory (or acoustics) to quantum theory is much easier than the step from point mechanics to quantum theory, because in electromagnetics you already deal with waves and modes of oscillation and solve eigenvalue equations in order to find the modes. You just have to treat a single electron like a wave, just in the same way as light is treated in classical electromagnetics.

In this picture, the only difference between classical physics and quantum physics is that in classical physics you can excite the modes of oscillation to a continuous degree, called the classical amplitude, while in quantum physics, the modes are "occupied" discretely. -- Fermionic modes can be occupied only once at a given time, while Bosonic modes can be occupied several times at once. Particles are just occupations of modes, no more, no less. As there are superpositions of modes in classical physics, you get in quantum mechanics quantum superpositions of occupations of modes and the scaling and phase-shifting factors are called (quantum) amplitudes. In a Carbon atom, for example, you have a combination of occupations of 6 electronic modes of low energy (i.e. frequency). Entangled states are just superpositions of combinations of occupations of modes. Even the states of quantum fields can be completely described in this way (except for hypothetical topological defects).

As you can choose different kinds of modes in acoustics and electromagnetics, for example plane waves, spherical harmonics or small wave packets, you can do so in quantum mechanics. The modes chosen will not always be decoupled, for example if you choose plane waves as the system of acoustic modes in the resonance corpus of a guitar, you will get reflexions on the walls of modes into different modes, i.e. you have coupled oscillators and you have to solve a coupled system of linear equations in order to describe the system. The same is done in quantum mechanics: different systems of eigenfunctions are just a new name for the same concept. Energy eigenfunctions are decoupled modes, while eigenfunctions of the position operator (delta-like wavepackets) or eigenfunctions of the angular momentum operator in a non-spherically symmetric system are usually strongly coupled.

What happens in a measurement depends on the interpretation: In the Copenhagen interpretation you need to postulate a collapse of the wavefunction to some eigenmode of the measurement operator, while in Everett's Many-worlds theory an entangled state, i.e. a superposition of occupations of modes of the observed system and the observing measurement apparatus, is formed.

The formalism of quantum mechanics and quantum field theoryEdit

In Dirac's formalism, superpositions of occupations of modes are designated as state vectors or states, written as |\psi\rangle (\psi being the name of the superposition), the single occupation of the mode n by |\psi_n\rangle or just |n\rangle. The vacuum state, i.e. the situation devoid of any occupations of modes, is written as |0\rangle. Since the superposition is a linear operation, i.e. it only involves multiplication by complex numbers and addition, as in

|\phi\rangle = \sqrt{1/3} \,|n\rangle - \sqrt{2/3} \,|m\rangle

(a superposition of the single occupations of mode n and mode m with the amplitudes \sqrt{1/3} and -\sqrt{2/3}, respectively), the states form a vector space (i.e. they are analogous to vectors in Cartesian coordinate systems). The operation of creating an occupation of a mode n is written as a generator \hat a^\dagger_n (for photons) and \hat b^\dagger_n (for electrons), and the destruction of the same occupation as a destructor \hat a_n and \hat b_n, respectively. A sequence of such operations is written from right to left (the order matters): In \hat a^\dagger_i \hat b^\dagger_m \hat b_n an occupation of the electronic mode n is moved to the electronic mode m and a new occupation of the electromagnetic mode i is created -- obviously, this reshuffling formula represents the emission of a photon by an electron changing its state. \alpha \, \hat a^\dagger_i \hat b^\dagger_m \hat b_n + \beta \, \hat a^\dagger_j \hat b^\dagger_m \hat b_n is the superposition of two such processes differing in the final mode of the photon (i versus j) with the amplitudes \alpha and \beta, respectively.

If the mode numbers are more complex -- e.g. in order to describe an electronic mode of a Hydrogen atom, (i.e. an orbital) you need the 4 mode numbers n, l, m, s -- the occupation of such a mode is written as |n, l, m, s\rangle or \hat b^\dagger_{n, l, m, s}|0\rangle (in words: the situation after creating an occupation of mode (n, l, m, s) in the vacuum). If you have two occupations of different orbitals, you might write

|n_1, l_1, m_1, s_1; n_2, l_2, m_2, s_2\rangle or \hat b^\dagger_{n_1, l_1, m_1, s_1}\hat b^\dagger_{n_2, l_2, m_2, s_2}|0\rangle.

It is important to distinguish such a double occupation of two modes from a superposition of single occupations of the same two modes, which is written as

\alpha \, |n_1, l_1, m_1, s_1\rangle + \beta \, |n_2, l_2, m_2, s_2\rangle or (\alpha \, \hat b^\dagger_{n_1, l_1, m_1, s_1} + \beta \, \hat b^\dagger_{n_2, l_2, m_2, s_2}) |0\rangle.

But superpositions of multiple occupations are also possible, even superpositions of situations with different numbers or different kinds of particles:

(\alpha \, \hat b^\dagger_{n_3, l_3, m_3, s_3} \hat b^\dagger_{n_1, l_1, m_1, s_1} + \beta \, \hat b^\dagger_{n_2, l_2, m_2, s_2} + \gamma \, \hat a^\dagger_i) |0\rangle.