Math Basics Math Menus Functions Variables Built-in Math Animation
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QM Waves Eigen-Energies Scattering QM Statistics QM Math
QMTools recognizes numerous functions and several variables of particular relevance to quantum applications. These are summarized below, along with any applicable usage notes.
QM Reserved Variables
QMTools recognizes two physical constants: ℏ (Planck's constant ÷ 2π) and ℏc (ℏ × lightspeed). Their numerical values depend upon the units chosen in the applet.
The value of these constants in the current system of units can be viewed
by entering the appropriate symbol(s) on the Math Palette (type
alt+K to insert ℏ), followed by ctrl+Z.
QM-Related Functions
In the table below, V identifies the system potential energy function, and m is the particle mass.To be properly interpreted, references to function values collectively must include the notation "(..)" following the function name.
| Description | Usage | Comments |
|---|---|---|
| fourier transform | fft(f(..), k) | transform variable k |
| inverse transform | fftinv(g(..), x) | g(k), f(x) are transform pair |
| average value | avg(f(x), w(..)) | weights |w(x)|2 |
| standard deviation | std(f(x), w(..)) | weights |w(x)|2 |
| cumulative weight | wts(w(..), a, b) | weights |w(x)|2 on [a,b] |
| kinetic energy | avgK(ψ(..), m) | coordinate wave ψ(x) |
| inner product | ovr(φ(..), ψ(..)) | matching args for φ, ψ; complex result |
| propagator | adv(V(..), m, φ(..), x, t) | V(x), initial wave φ(x) |
| spectral filter | spf(V(..), m, a(E)) | envelope function a(E) |
| spectral transform | spt(a(..), x, t) | spectrum a(E); 't' argument optional |
| stationary wave¶ | sta(V(..), m, E, x) sta(V(..), m, E, ell, r) |
1d or 3d (radial) wave |
| stationary wave 3d | sta3(g(..), em, x, y, z) | radial wave g(r); integer em (magnetic #) |
| incident wave | inc(ψ(..), x) | scattering wave ψ(x) |
| reflected wave | ref(ψ(..), x) | scattering wave ψ(x) |
| reflectance | refl(V(..), m, E) | energy E |
| transmittance | trns(V(..), m, E) | energy E |
| phase shift | phase(V(..), m, E, ell) | energy E; integer ell (orbital #) |
¶This function has several variants: the 5-argument form finds radial waves for spherically symmetric potentials, and requires the orbital quantum number as the 4th argument, followed by the coordinate label as the 5th argument. All versions also accept an optional time parameter as the last argument in the list.