Python Calculator

1. Units, Constants and quick calculations¶

scipy.constants knows every physical constant (full list):

A scratchpad, seeded with a photon-energy calculation. Edit it and press play; the value of the last line is displayed:

2. Plotting panel¶

A ready-to-edit plotting template. Swap in any function of x; the slider is yours to repurpose:

3. Numerical integral¶

Define any function and limits in the pad; the plot shades area above the axis in green and below in red, and the value comes from np.trapezoid, the same tool your homework uses:

The green and red areas fight each other: an integral is a signed sum. For probability densities like $|\psi|^2$ the red never appears, which is exactly why they can be interpreted as probabilities.

4. Symbolic window¶

Derivatives, integrals, and equation solving with sympy; results render as typeset math. Each pad is editable:

5. Orbital visualizer¶

Pick quantum numbers (the menus only ever offer valid combinations). The first plot is a 2D cross-section of the orbital in the xz-plane, with blue and red marking the wavefunction’s positive and negative lobes. Below it, the radial view shows the radial function $R_{nl}(r)$ and the radial distribution $P(r) = r^2 R^2$, and reports the most probable radius, the mean radius $\langle r\rangle$, and the node count.

6. One-dimensional Schrödinger solver¶

Pick a potential and get its bound states instantly: energies as horizontal lines, wavefunctions drawn at their own energy (units: $\hbar = m = 1$, hard walls at $x = \pm 4$). To see how the solver works inside, open the numerical Schrödinger lab.

Try the classics: the box gives the $n^2$ ladder, the harmonic well gives perfectly even spacing (the fingerprint of vibrations), the linear well spaces levels like Airy zeros, and the double well pairs levels into tunneling doublets. The square barrier does the same with sharp walls: it cuts the box into two halves that talk only by tunneling, so raise $a$ and watch each pair of levels squeeze together.

7. Tunneling through a square barrier¶

An electron with energy $E$ comes in from the left toward a barrier of height $V_0$ and width $a$. Classically it bounces back whenever $E < V_0$. Quantum mechanically the wave leaks in, decays as $e^{-\kappa x}$ inside, and whatever amplitude reaches the far side carries on as a transmitted wave. The fraction that gets through is

For a thick barrier ($\kappa a \gg 1$) this becomes $T \approx 16\frac{E}{V_0}\left(1 - \frac{E}{V_0}\right)e^{-2\kappa a}$, exponential in the width. Units here are eV and Å with the electron mass. The derivation is in the tunneling lecture.

Start below the barrier and widen it in steps of 1 Å: $T$ drops by roughly the same factor each step, and the log scale turns that exponential into a straight line. Then push $E$ above $V_0$: the classical electron always passes, but the quantum one still reflects, and $T$ wiggles up to 1 only at the resonances. The same exponential is why a scanning tunneling microscope current changes about tenfold per ångström of tip height, and why a proton, 1836 times heavier than an electron, has a $\kappa$ about 43 times larger and tunnels only through barriers a fraction of an ångström thick.

8. Quadratic equation solver¶

Every quadratic $ax^2 + bx + c = 0$ is solved by one formula:

The sign of the discriminant $\Delta = b^2 - 4ac$ decides what comes out: two real crossings for $\Delta > 0$, one repeated root for $\Delta = 0$, and for $\Delta < 0$ a pair of complex conjugates that never touch the axis. Drag the coefficients and watch the roots leave the real line.

Start from the defaults and raise $c$: the two crossings slide together, merge at $\Delta = 0$, and then step off the real line as a mirror pair, real part shared and imaginary parts opposite. The curve stops touching the axis, but the number of roots never changes. That is the fundamental theorem of algebra in miniature: a degree $n$ polynomial has exactly $n$ complex roots. The background is in the complex numbers appendix.