Symbolic Math with SymPy

SymPy is the SciPy ecosystem’s library for symbolic mathematics: it manipulates expressions exactly, like a free Mathematica built on Python. Where NumPy gives you numbers, SymPy gives you formulas: it can expand, factor, differentiate, integrate, and solve algebraically. Before a variable can appear in an expression it must be declared a symbol with symbols().

Algebra¶

The everyday manipulations are expand, factor, and simplify.

To solve an equation (SymPy sets the expression equal to zero), use solve:

Calculus¶

Differentiate with diff: the first argument is the expression, the rest are the variables to differentiate by. Repeating a variable (or giving a number) takes higher derivatives.

Integrate with integrate. With no limits you get an antiderivative; with limits, a definite integral. The symbol oo means infinity, so the Gaussian integral central to quantum mechanics is one line:

A Taylor series expansion uses series. By default it expands about $x = 0$; extra arguments set the center and the order.

A quantum-mechanical example: hydrogen radial functions¶

SymPy ships the hydrogen atom’s exact wavefunctions in its physics module. Here we pull the $2p$ radial function $R_{21}(r)$, find the radius where the radial probability density $r^2 R_{21}^2$ peaks, and plot it. This is a worked symbolic-to-numeric pipeline: solve exactly, then lambdify into a NumPy function to plot.

The peak of the $2p$ radial density sits at $r = 4\,a_0$, exactly the Bohr-model radius of the $n = 2$ orbit.

Exercises¶

1. Factor $x^2 + x - 6$.

2. Expand $(x - 2)(x + 5)\,x$.

3. Evaluate $\displaystyle\int_0^{\ln 4} \frac{e^x}{\sqrt{e^{2x} + 9}}\,dx$ with integrate (use log(4) for the upper limit).

4. The reversible isothermal work of a gas is $w = \int_{V_i}^{V_f} -\frac{nRT}{V}\,dV$. Integrate it symbolically, then evaluate for $n = 2.44$ mol, $T = 298$ K, $V_i = 0.552$ L, $V_f = 1.32$ L.

Try it live¶