Studies in Composing Hydrogen Atom Wavefunctions
Lance J. Putnam,^{1}, JoAnn KucheraMorin^{2}, Luca Peliti^{3, 4}
^{1}(Academic, composer), Department of Architecture, Design and Media Technology, Aalborg University, Sofiendalsvej 11, DK9200 Aalborg (Denmark). E − mail:
lp@create.aau.dk
^{2}(Academic, composer), Media Arts and Technology, University of California, Santa Barbara, Santa Barbara, CA, 931066065 (USA). E − mail:
jkm@create.ucsb.edu
^{3}(Academic), Dipartimento di Fisica, Università “Federico II”, Complesso Monte S. Angelo, 80126 Napoli (Italy). E − mail:
peliti@na.infn.it
^{4}INFN, Sezione di Napoli, Complesso Monte S. Angelo, 80126 Napoli (Italy)
We present our studies in composing elementary wavefunctions of a hydrogenlike atom and identify several relationships between physical phenomena and musical composition that helped guide the process. The hydrogenlike atom accurately describes some of the fundamental quantum mechanical phenomena of nature and supplies the composer with a set of welldefined mathematical constraints that can create a wide variety of complex spatiotemporal patterns. We explore the visual appearance of timedependent combinations of two and three eigenfunctions of an electron with spin in a hydrogenlike atom, highlighting the resulting symmetries and symmetry changes.
1 Introduction
We attempt to represent the state of a quantum system and its evolution in an immersive integrated environment, as a way to obtain, for the scientist, a better intuitive understanding of quantum reality and to produce, for the composer, a powerful medium for aesthetic investigations and creative expression. Specifically, the hydrogenlike atom is important for scientists as it accurately describes some of the fundamental quantum mechanical phenomena of nature. For the composer, it supplies a set of welldefined mathematical constraints that can be adjusted to create a wide variety of spatiotemporal structures.
Making such a quantum system perceptible is of utility to both the scientist and artist. First of all, doing so renders the abstract mathematical system into a format that can be experienced in a way that is more intuitive and visceral. Such a representation will not necessarily contribute to our understanding of the Hilbert space in which the atom lives, but can give a feeling of this abstract space in the more familiar space we all live. Second, perceptualizing the atom can assist in understanding emergent patterns, flows, symmetries and dynamics when multiple hydrogenlike wavefunctions are mixed in superposition. It can be difficult to represent these complex phenomena in one’s head from the mathematical equations alone.
In order to represent the wavefunctions, we use the AlloSystem software framework
[1] to facilitate the interactive visualization of this information and the AlloSphere
[2], a threestory, sphericallyshaped virtual reality environment, to allow full immersion in the data. The AlloSphere not only can provide an intuitive understanding of the hydrogenlike atom through perception
[14], but also permits observation of multiple levels of structure in a way that is more natural than a flat display (wall) (Figure
1↓). The hydrogenlike atom is a test bed for the AlloSystem software and AlloSphere instrument and a necessary prerequisite towards representing more complex and nuanced quantum systems. We decided, as a first step, to represent the wavefunctions visually as this appears to be more easily understood than, for instance, translating them into sound.
2 Related Work
Over the past century, many approaches have been taken to visualizing atomic orbitals, mostly for didactical purposes. The first visualizations of hydrogen atom orbitals were made in 1931 using time exposure photography of the motions of a modified mechanical lathe
[7]. Berndt Thaller has done extensive work on the visualization of quantum mechanical states, in particular in threedimensional systems
[4, 3] and hydrogenlike atoms, by means of several visualization techniques. However, the stress laid on the temporal evolution of the wavefunctions is comparatively limited.
Several interactive tools have been made that visualize dynamic mixtures of hydrogenlike atom wavefunctions in realtime. Falstad’s
Hydrogen Atom Applet [17] displays a volume rendering of mixtures of any number of orbitals, but only with low energy levels. Dauger’s
Atom in a Box [6] also employs volume rendering and allows mixing of up to eight wavefunctions with relatively high energy levels. These tools, however, do not represent wavefunctions with spin.
The time evolution of an arbitrary initial wavefunction is visualized by a software package developed by Belloni and Christian
[16, 15]. The method applies to onedimensional system with a purely discrete spectrum. The wavefunctions are projected on the eigenfunctions of the system and the subsequent evolution is obtained by evolving the coefficients of the decomposition. While the subtleties of quantummechanical evolutions are well captured by this software, its application to higherdimensional systems would be quite difficult.
Artistically, we identify our work closely with John Whitney’s concept of digital harmony
[11] in applying harmonics and other musical notions, such as scales and transitions of curves, to visual composition. Julian VossAndreae’s quantum sculptures
[12] are also relevant being visual interpretations of quantum mechanics, however, are more conceptual in nature and do not give a picture of the complex dynamics of quantum systems.
3 Conceptual Framework
Our conceptual framework is based on a particular process of musical composition, namely, the organization of frequencies and pitches, transposed from the sound domain to the visual domain. More specifically, we propose working with wavefunctions in a similar way to how a composer works with musical tones in creating a larger work.
Quantum mechanics dictates that the evolution of a quantum state is realized by a process that closely resembles additive synthesis in music. A general timedependent quantum state is obtained by the superposition of a number of wavefunctions representing stationary states, the eigenfunctions, multiplied by periodically varying coefficients. The eigenfunctions correspond to the simple sounds that add together to form the complete synthesized sound. Their superposition produces the complex spatiotemporal pattern described by the timedependent wavefunction.
In the following, we present several concepts in musical composition and physics that support our analogy between wavefunctions and musical tones.
3.1 Compositional Process
Music carries meaning on several timescales, from individual timbres and pitches (foreground) to short melodies and rhythms (middleground) all the way up to largescale form and structure of a work (background), each engaging distinct perceptual and cognitive processes. Typically, when composers start to realize a piece of music they either have a big idea at the macrolevel (a large structure that they want to unfold), or they start with something at the microlevel such as a small group of notes. One strategy composers use to master their data in unfolding a piece is to work at a middleground level, that is, controlling the larger structure at the background while unfolding the microstructure at the foreground
[8]. Also, in predicting local motion in selfgenerative music from the microstructure, a middlegroundtoforeground approach may aid in local directional choices. In Xenakis’ compositional approach, local events are decided randomly, however these events are constrained by clearly articulated probability distributions
[10]. This approach is similar to the relationship between the deterministic Schrödinger evolution and probabilistic measurement process in quantum mechanics.
Sketching is a technique used in composition where a composer will try out various scenarios of harmonic structures in creating a work. Typically a composer will write a series of studies called études, that are part of the sketching process. The sketching process is hierarchical in that a rough draft of the structure is sketched, and then various parts are filled in at the microlevel, which will cause adjustments at the macrolevel. This is the middleground area that functions as the pendulum between the micro and macro layers. Since this process is hierarchical, it can be applied at the level of adding frequencies together to build larger structures.
Our initial approach in composing with the hydrogenlike atom is to start with a middleground which we identify as interference patterns. Thus, we can experiment with various combinations of multiple wavefunctions that are scientifically correct and we find aesthetically pleasing. We begin the process with sketchinga series of studies mixing multiple eigenfunctions together. The studies range from the physicallymotivated to the openended, exploring beating and complex interference patterns such as finesplitting. Thus, we are creating the basic musical alphabet by mixing the multiple eigenfunctions together with certain chosen characteristic frequencies.
3.2 Eigenfunctions
We employ eigenfunctions as our basic units of composition. Eigenfunctions are identified by a set of quantum numbers that determine the particular shape of the eigenfunction in space (Section
3.5↓) as well as its frequency of evolution through time. Controlling the effect of the quantum numbers on the visual pattern that is produced is the basic step in the creation of a satisfactory composition.
A quantum state represented by a single eigenfunction evolves through a spaceindependent phase factor that changes periodically in time with its characteristic frequency. Thus, the spatial distribution of an eigenfunction’s magnitude does not change with time. Additionally, two different eigenfunctions can share the same characteristic frequency. An eigenfunction is analogous to a normal mode in a pipe while a characteristic frequency is analogous to a pure tone.
Eigenfunctions can be placed in superposition to create a wavefunction (see Appendix). Such wavefunctions generally have more complex properties than their constituent eigenfunctions. This is analogous to how more complex tones can be built from simple tones, as done with an organ, or from musical voicing and chorusing. These emergent complexities can often be identified with interference patterns.
3.3 Interference Patterns
From an artistic standpoint, the phenomenon of interference is interesting as it allows complex emergent patterns to be constructed from an “alphabet” of simple parts. Strong interference patterns occur when two or more waveforms with similar frequencies are superposed.
An elementary form of interference, beating, occurs when two sinusoids with similar frequencies are summed together.Musically speaking, beating effects can enrich static tones by imposing an evolution of timbre relative to pitch that is an order of magnitude slower. To obtain a complex beating pattern, one can mix two waveforms with the same timbre, but slightly different pitch. This technique called celeste was devised by early organ builders to enrich the sound of individual notes. Similarly, mixing two eigenfunctions produces a spatial pattern that evolves periodically in time, with a frequency equal to the difference between the characteristic frequencies of the eigenfunctions.
While celeste produces a more complex time evolution of a waveform, the effect is strictly periodic in both the time and frequency domains. To alleviate these periodicities, one must mix three or more waveforms together. A wavefunction obtained by superposing three or more eigenfunctions produces a pattern that in general never repeats itself. Interestingly, finesplitting in an atom can be represented with a mixture of three eigenfunctions.
3.4 Hydrogenlike Atom
A hydrogenlike atom consists of a single particle of mass
m subject to a central force whose potential is inversely proportional to the distance from the nucleus. The characteristic frequencies given by this spectrum rapidly become smaller, leading to slower evolution, as the principal quantum number
n increases. They depend only on
n and
J, and the
J dependence (a spinorbit coupling effect) is very weak. The
J dependence produces the finesplitting of spectral lines, discussed in section
4.3↓.
The hydrogen eigenfunctions provide sufficient complexity to be used as building blocks for our composition. Using hydrogen rather than, say, the harmonic oscillator eigenfunctions is analogous to using an organ to create music rather than individual sine waves. This lends sufficient interest in using the compositional process of mixing waveforms to build chords that act as building blocks for visualization. However, the fact that a great number of eigenfunctions exhibit the same characteristic frequency strongly reduces the possibilities to obtain beating patterns. We took therefore the liberty to assign frequencies to the eigenfunctions in an arbitrary way.
3.5 Geometric Properties of the Eigenfunctions
To assist in composition, it is useful to understand the geometrical properties of the eigenfunctions. It is convenient to start from the eigenfunctions of the spinless electron identified by the quantum numbers
(n, ℓ, m). They are characterized by a number of windings, shells and stacks, i.e., elementary patterns relative to a sphere with a north and south pole on the
zaxis (Table
1↓). Windings are the cycles made by the phase of the eigenfunction in making one loop around the
zaxis, stacks are the parallel slices running perpendicular to the
zaxis, and shells are the concentric regions separated by spherical surfaces where the eigenfunction vanishes.
Parameter



Range

Winding number

n_{w}

: = m

…, − 2, − 1, 0, 1, 2, …

Number of stacks

n_{s}

: = ℓ − m + 1

1, 2, …

Number of shells

n_{r}

: = n − ℓ

1, 2, …

Table 1 Geometric properties of spinless eigenfunctions in terms of quantum numbers.
The eigenfunctions of the electron with spin are obtained by superimposing two spinless eigenfunctions with the same values of n and ℓ, but values of m differing by 1. Thus, one eigenfunction has one more winding and one less stack than the other. As the spin eigenfunction is identified by a fourdimensional spinor at each point in space, rather than a complex number as in the spinless version, it cannot be simply described in terms of windings and stacks. However, in general, higher quantum numbers correspond to more spatially extended and varying eigenfunctions.
4 Studies
Our studies aimed to explore the emergent structures and dynamics resulting from mixing together multiple eigenfunctions. Due to the vast number of possible eigenfunction combinations, we decided to limit our scope, for now, to mixtures of only two or three eigenfunctions.
To visualize a wavefunction with spin, we use a colormapped isosurface as we find it gives a clear picture of the wavefunction’s overall magnitude and phases. The surface is drawn at a particular value of the wavefunction magnitude
Ψ. The surface is colored according to the spin vector phases
θ (spinup/down state) and
φ (relative phase). We map
θ to hues going between orange (spin up) and cyan (spin down) and
φ to a grayscale gradient. The grayscale gradient is discontinuous revealing an abrupt cut where the phase completes the cycle
[9]. The cuts are more indicative than a smooth mapping and helpful in distinguishing between the two different phases. The hues and grays are interpolated between according to the proximity of the spin vector to the equator; more hue near the poles and more gray near the equator (Figure
2↓).
4.1 One Eigenfunction
There are already a number of programs allowing for the visualization of single eigenfunctions of the hydrogenlike atom
[18][5]. Since the time dependence of a single eigenfunction only appears in an overall phase factor, it leads to a completely timeindependent image in our representation, where the overall phase factor is neglected. Thus the representation of single eigenfunctions lies outside the scope of the compositional process, and we did not investigate it systematically. Indeed, a number of software projects have already tackled the problem of visualizing the eigenfunction of the hydrogenlike atom, both in its spinless version and in the one with spin
[4, 3]. We are more interested in the timedependent patterns one obtains by mixing two or more eigenfunctions.
4.2 Two Eigenfunctions
4.2.1 General Patterns
When mixing two eigenfunctions, we already observe a wide variety of patterns. The patterns evolve periodically either by changing shape or by rotating, with a frequency proportional to the energy difference of the eigenstates. In order to make this evolution visible, we arbitrarily lower this frequency. Fortunately, these patterns can be classified into a handful of general categories of shapes and dynamics that can be related to the quantum numbers.
We observed the following mutually exclusive types of dynamics with respect to the wavefunction magnitude and spin:

No change over time.

Rotation around the zaxis (with possible exception of relative phase).

Beating of magnitude

isotropically,

radially, and

along the zaxis.
Type 1 dynamics involve no change in the wavefunction magnitude and spin vector. This occurs whenever the characteristic frequencies
ϵ_{ν} are equal. Type 2 dynamics display a periodic rotation of the wavefunction around the
zaxis. There is also a periodic change in the relative phase of the spin at the same rate. One can interpret this rotation as an angular traveling wave. Type 3 dynamics are the most interesting and consist of beating effects that may involve a change in the shape of the wavefunction magnitude. The primary constraint on type 3 dynamics is that the
js are equal. Beating patterns are rotationally invariant around the
zaxis. For type 2 and 3 dynamics, the rate of change of the wavefunction magnitude is equal to the difference between the two characteristic frequencies. The exact constraints on the eigenfunction parameters are given in Table
2↓.
Case

n

ℓ

J

j

ϵ_{ν}

Comments

1





=


2




≠

≠


3(a)

=

=

=

=

≠


3(b)

≠

=

=

=

≠


3(c)


ℓ_{1} + ℓ_{2} odd


=

≠

Asymmetric w.r.t xyplane

3(c)


ℓ_{1} + ℓ_{2} even

≠

=

≠

Symmetric w.r.t xyplane

Table 2 Parameter constraints for twoeigenfunction dynamic types. Empty cells indicate no dependence on the parameter.
The shape of the wavefunction magnitude is limited to a few particular types of point group symmetries. The difference between the js is the order of dihedral symmetry around the zaxis, i.e., the number of planes of reflection passing through the zaxis. When the js are equal, there are an infinite number of reflection planes and thus the shape is rotationally invariant around the zaxis. For nonbeating wavefunctions, the shape exhibits either reflection or rotationreflection symmetry with respect to the xyplane. The rules are
ℓ_{1} + ℓ_{2} + j_{1} + j_{2} → ⎧⎨⎩
even
rotationreflection
odd
reflection
Changing
J does not effect symmetry type since it only changes the eigenfunction weights. Complex wavefunctions displaying the two types of symmetry are shown in Figure
3↓.
4.2.2 Lightemitting Combinations
Although we cannot represent the process of light emission in our system, we can illustrate it through a superposition of eigenfunctions which satisfy the selection rules (see Appendix). This leads to a number of interesting wavefunctions which periodically change at a rate equal to the frequency of the emitted photon. Figure
4↓ shows all such superpositions which can be built out of the lowest values of
n. Interestingly, they include all the nontrivial types of dynamics and three distinct types of spin distributions.
When the
n values are very different, one sees a relatively smaller radial interference. Indeed, most of the interference takes place in the region occupied by the eigenfunction with the smaller
n value. More complex patterns can be found when the
n values are close and larger (Figure
5↓).
It appears that the only shapes for light emission superpositions are rotationally invariant around z or have reflective symmetry with respect to the xy plane. None exhibit rotationreflection symmetry.
4.3 Three Eigenfunctions
Combining three eigenfunctions, each with its own frequency, yields in general a beating pattern which never reproduces itself exactly. The number of possible combinations rapidly increases, and we did not yet perform a systematic study of the resulting patterns. As with the simple combinations, the more interesting wavefunctions resulted by choosing values of n close to one another, since in this case the spatial extension of the combining eigenfunctions have a larger overlap.
A special case of physical interest is an illustration of finesplitting of an emission spectral line. This corresponds to preparing the system in a superposition of three eigenfunctions: a lowenergy eigenfunction and two ‘‘excited’’ eigenfunctions (details in Appendix). The resulting wavefunction exhibits a strong beating with a carrier frequency roughly equal to the difference between the frequencies of the excited and the lowlying eigenfunctions, slowly modulated with a frequency equal to the difference between the frequencies of the two excited eigenfunctions. Such a combination is shown in Figure
6↓. In a real physical situation the modulation is at least about 18,800 times slower than the beating frequency. Thus, if we were to rescale the involved frequencies faithfully, the modulation would be too slow to be visible. We chose therefore to rescale the frequencies independently, keeping the modulation frequency somewhat smaller than the fundamental one.
One can explore different combinations of three eigenfunctions in a similar way. Figure
7↓ shows a combination of higherquantum number eigenfunctions, which exhibits a slow modulation of the wavefunction shape superimposed on a global rotation. These combinations would apply, e.g., to an atom immersed in a magnetic field directed along the
zaxis.
5 Future Work
5.1 Probability Flow
The evolving shape of the probability distribution is connected with the probability density flow, which demands to be visualized itself. One could achieve this by a static representation of the velocity field v(r, t), or, more intuitively, by representing via agents the actual motion with the local velocity. We have explored this option by introducing a small number of agents moving with the local velocity. The dynamics of these agents is connected to the changing shape of the wavefunction in a nontrivial and intriguing way, but a more systematic study is needed.
5.2 Timedependent Perturbation as a Compositional Process
In order to introduce a more complex temporal behaviorlike going from a fixed chord to a melody which evolves in timeone could add a timedependent perturbation V_{I}(t) to the Hamiltonian. This perturbation changes the wavefunction from, say, a single eigenfunction to a complex superposition of many eigenfunctions. Thus one could use in principle a wellchosen perturbation to produce a number of different wavefunctions in succession.
There are a number of technical difficulties in following this path. On the one hand, one would need enough computer power to solve the fullfledged timedependent Schrödinger equation. One could however make the task more manageable by projecting back the evolution onto a chosen finitedimensional subspace of wavefunctions, what allows us to exploit the already implemented representation of the wavefunctions. On the other hand, it is hard to identify which perturbation leads to a given sequence of wavefunctions. While the timedependence of weights and phases could be chosen arbitrarily, it could be interesting to identify physically relevant perturbations which lead to interesting pattern sequences.
6 Conclusion
With the present representation of the hydrogenlike atom, we were able to study various wavefunction combinations that form “chords” from our quantum alphabet of musical “tones”. We worked together on a daily basis, eventually arriving at the current mapping representations. We implemented different iterations of the mappings, most of which were superseded as either not informative enough from the physical point of view, or not aesthetically pleasing. For example, we employed pointcloud and vectorfield representations of the wavefunction, however, the visual pattern was dominated by the distribution of these reporter glyphs
[13]. The isosurface representation has the advantage of leading to an immediate grasp of the broad features of the probability density distribution described by the wavefunction. The subtler effects contained in the spinor amplitudes and phases are reasonably well reported via our color coding. We found with some satisfaction that physicallybased combinations often lead to aesthetically pleasing patterns, as in the case of the lightemitting superposition.
It is easy to use the system for a visual investigation of the properties of the hydrogen atom eigenfunctions, as a function of their quantum numbers. Thus the system can be efficiently exploited as a didactic tool. However, more insight is gained by looking at the unfolding of the visual patterns of a combination of several eigenfunctions. This behavior also provides a more intuitive understanding of the mechanisms of timedependent perturbation theory.
Appendix
A hydrogenlike atom wavefunction
Ψ_{ν}(r) is given as a product of independent radial and angular functions
Ψ_{ν}(r) = ⎧⎨⎩
R_{n}(r)Y_{ℓm}(θ, φ),
if spin is neglected,
R_{n}(r)ψ_{ℓJj}(θ, φ, s),
with the spin
where

For the spinless electron: ν = (n, ℓ, m);

For the electron with spin: ν = (n, ℓ, J, j).
Here (r, θ, φ) are the spherical coordinates of the position r, s = ↑, ↓ is the spin quantum number, and

n is the principal quantum number: n = 1, 2, 3, …;

ℓ is the angular quantum number: ℓ = 0, 1, …, n − 1;

m is the magnetic quantum number: m = − ℓ, − ℓ + 1, …, ℓ − 1, ℓ;

J is the total angular momentum: J = ℓ±(1)/(2);

j is the zcomponent of the total angular momentum: j = − J, − J + 1, …, J − 1, J.
The wavefunctions satisfy the timedependent Schrödinger equation
iℏ∂_{t}Ψ = ĤΨ
where the Hamiltonian operator
Ĥ corresponds to the energy of the system. The eigenfunctions
Ψ_{ν} are solutions of the time
independent Schrödinger equation
ĤΨ_{ν} = ϵ_{ν}Ψ_{ν}
and are identified by a collection
ν of quantum numbers. A general timedependent wavefunction
Ψ is obtained as a linear combination of eigenfunctions with timevarying coefficients
c_{ν}(t) which evolve according to
c_{ν}(t) = e^{iϵνt ⁄ ℏ}c_{ν}(0).
While the amplitude
Ψ^{2} = Ψ_{↑}^{2} + Ψ_{↓}^{2} yields the probability of finding the electron at a given position in space, there is much physical information contained in the full
Ψ, e.g., the probability of observing a given value of the spin.
We only consider in this work pure states, which are described by a complexvalued wavefunction Ψ(x). In our case x = (r, s), so Ψ can also be considered a spinor, i.e., a twocomponent vector Ψ = (Ψ_{↑}, Ψ_{↓}). Each component of the spinor wavefunction is proportional to an eigenfunction of the spinless electron with a proportionality coefficient dictated by symmetry. Thus each spinor eigenfunction has two components, each of which is an eigenfunction of the spinless electron with different values of m. Given j, the upper and lower eigenfunctions have m = j − (1)/(2) and m = j + (1)/(2), respectively. One can associate to each spinor a 3D spin vector v which points in the local direction of the spin.
Light is emitted when the electron, coupled to an electromagnetic field, performs a transition from a state with quantum numbers ν and energy ϵ_{ν} to a state with quantum numbers ν’ and a smaller energy ϵ_{ν’} (the reverse transition corresponds to light absorption). The energy of the emitted photon is given by ϵ = ϵ_{ν} − ϵ_{ν’} and is related to its frequency ω by the Planck relation ϵ = ℏω. Symmetry dictates that the transition can only take place if ν and ν’ satisfy the selection rules:

For the spinless electron ν = (n, ℓ, m), ν’ = (n’, ℓ’, m’), with ℓ − ℓ’ = ±1, m − m’ = 0, ±1.

For the electron with spin, ν = (n, ℓ, J, j), ν’ = (n’, ℓ’, J’, j’), with ℓ − ℓ’ = ±1, j − j’ = 0, ±1.
Finesplitting of an emission spectral line corresponds to a superposition of three eigenfunctions: a lowlying eigenfunction with quantum numbers (n_{0}, ℓ_{0}, J_{0}), and two ‘‘excited’’ eigenfunctions with n_{1, 2} = n_{0}, ℓ_{1, 2} = ℓ_{0} + 1 and two different values J_{1} and J_{2} satisfying J_{1, 2} = ℓ±(1)/(2).
Acknowledgments
This material is based upon work supported by the National Science Foundation under Grant Numbers 0821858, 0855279, and IIS1047678.
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Glossary
Eigenfunction In quantum mechanics, a wavefunction associated with a specific possible value of a physical observable. A general wavefunction can be obtained as a superposition of eigenfunctions.
Perceptualization The mapping of data so that it can be perceived in a meaningful way through human senses, e.g., visualization and sonification.
Pure state A state of a quantum system which contains its fullest specification compatible with the laws of quantum mechanics. It is described by a wavefunction.
Schrödinger equation The equation describing the evolution of a quantum state. It reads iℏ∂_{t}Ψ = Ĥ Ψ, where Ĥ is the Hamiltonian operator associated with the energy of the system.
Spin A purely quantum degree of freedom of elementary particles, representing an intrinsic contribution to the total angular momentum. For electrons, the spin assumes the values ±ℏ ⁄ 2, where ℏ is the reduced Planck constant.
Superposition An additive mixture of waveforms, typically from the same family of functions. The parts cannot necessarily be recovered fully from the whole. The solutions of the Schrödinger equation are a superposition of eigenfunctions with timevarying coefficients.
Wavefunction The complexvalued function Ψ(x), defined on the configuration space of a quantum system, which contains a full description of a pure quantum state. Its evolution is described by the timedependent Schrödinger equation.