Calculus Essentials

Calculus is the language of change. In quantum mechanics we differentiate wavefunctions to build momentum and energy operators, and we integrate them to compute probabilities and expectation values. This page collects the essentials, with a picture for the two ideas that matter most: the derivative as a slope, and the integral as an area.

Differentiation¶

The derivative as a slope¶

The derivative of $f$ at a point is the slope of the line tangent to the curve there. We build it from the slope of a secant line through two nearby points and then let those points merge:

As the spacing $h$ shrinks, the secant line pivots into the tangent line. The plot below shows this for $f(x) = x^2$ at $x = 1$: three secant lines with shrinking $h$ close in on the true tangent of slope 2.

Source

import numpy as np
import matplotlib.pyplot as plt

f = lambda x: x**2
x0 = 1.0
x = np.linspace(-0.5, 2.5, 400)

fig, ax = plt.subplots(figsize=(6.5, 5))
ax.plot(x, f(x), 'k', lw=2, label=r'$f(x)=x^2$')

# Secant lines for shrinking h
for h, color in zip([1.0, 0.5, 0.2], ['#d1495b', '#edae49', '#66a182']):
    slope = (f(x0 + h) - f(x0)) / h
    sec = f(x0) + slope * (x - x0)
    ax.plot(x, sec, '--', color=color, lw=1.5, label=f'secant, h={h} (slope {slope:.1f})')
    ax.plot([x0, x0 + h], [f(x0), f(x0 + h)], 'o', color=color, ms=5)

# True tangent (slope 2)
ax.plot(x, f(x0) + 2 * (x - x0), color='#2e4057', lw=2, label='tangent (slope 2)')
ax.plot(x0, f(x0), 'ko', ms=6)

ax.set_xlim(-0.5, 2.5)
ax.set_ylim(-1, 6)
ax.set_xlabel('x')
ax.set_ylabel('f(x)')
ax.legend(fontsize=8, loc='upper left')
ax.set_title('Fig.1 Secant lines converging to the tangent as h shrinks')
plt.tight_layout()
plt.show()

The derivative is also the instantaneous rate of change: if $f$ is position, $f'$ is velocity; if $f$ is concentration, $f'$ is reaction rate. A function is differentiable only where it is smooth, so no corners, jumps, or vertical tangents.

Rules of differentiation¶

Table of common derivatives¶

A few limits underlie these results: $\displaystyle\lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1$, $\displaystyle\lim_{\theta\to 0}\frac{\cos\theta-1}{\theta}=0$, and the definition $\displaystyle\lim_{h\to 0}\frac{e^{h}-1}{h}=1$, which is exactly what makes $e^{x}$ its own derivative.

Integration¶

The integral as accumulated area¶

A definite integral measures the net area between a curve and the $x$ axis. We approximate that area with a Riemann sum of $n$ rectangles and let $n\to\infty$:

The plot shows the rectangles filling the area under $f(x)=x^2$ on $[0,2]$ as $n$ grows from coarse to fine. The exact area is $\int_0^2 x^2\,dx = 8/3 \approx 2.667$.

Source

import numpy as np
import matplotlib.pyplot as plt

f = lambda x: x**2
a, b = 0, 2
xs = np.linspace(a, b, 400)

fig, axes = plt.subplots(1, 3, figsize=(11, 3.6), sharey=True)
for ax, n in zip(axes, [4, 8, 32]):
    edges = np.linspace(a, b, n + 1)
    mids = 0.5 * (edges[:-1] + edges[1:])   # midpoint rule
    dx = (b - a) / n
    approx = np.sum(f(mids) * dx)
    ax.bar(mids, f(mids), width=dx, align='center',
           color='#66a182', edgecolor='white', alpha=0.7)
    ax.plot(xs, f(xs), 'k', lw=2)
    ax.set_title(f'n = {n},  sum = {approx:.3f}', fontsize=10)
    ax.set_xlabel('x')
axes[0].set_ylabel('f(x)')
plt.suptitle('Fig.2 Riemann sums approaching the exact area 8/3 = 2.667 as n grows', y=1.04)
plt.tight_layout()
plt.show()

We can watch that convergence numerically:

import numpy as np

f = lambda x: x**2
a, b = 0.0, 2.0
for n in [4, 16, 64, 256, 1024]:
    x = np.linspace(a, b, n + 1)
    mids = 0.5 * (x[:-1] + x[1:])
    approx = np.sum(f(mids) * (b - a) / n)
    print(f"n = {n:5d}   Riemann sum = {approx:.5f}")

exact = 8 / 3
print(f"\nexact value  = {exact:.5f}")

n =     4   Riemann sum = 2.62500
n =    16   Riemann sum = 2.66406
n =    64   Riemann sum = 2.66650
n =   256   Riemann sum = 2.66666
n =  1024   Riemann sum = 2.66667

exact value  = 2.66667

Antiderivatives and the fundamental theorem¶

An antiderivative $F$ of $f$ satisfies $F'(x)=f(x)$. Because the derivative of a constant is zero, antiderivatives come as a family $F(x)+C$. The fundamental theorem of calculus ties the two halves of the subject together:

This is why we rarely compute Riemann sums by hand: finding an antiderivative turns an infinite sum into a single subtraction.

Table of indefinite integrals¶

Useful properties: reversing the limits flips the sign, $\int_a^a f\,dx = 0$, and integrals are linear, $\int (cf \pm g)\,dx = c\int f\,dx \pm \int g\,dx$. For symmetric limits, an even function gives $\int_{-a}^{a} f\,dx = 2\int_0^a f\,dx$ while an odd function gives $\int_{-a}^{a} f\,dx = 0$, a shortcut that kills many quantum-mechanical integrals on sight.

Techniques of integration¶

Integration by parts is the reverse of the product rule, and it is the standard tool for integrals like $\int x\,e^{-x}\,dx$ or $\int x\sin x\,dx$ that appear constantly in expectation-value calculations.

For products of sines and cosines, the power-reduction and product-to-sum identities do the work:

These are exactly the integrals that enforce orthogonality of particle-in-a-box wavefunctions.

Problems¶

Problem 1: Differentiate a Gaussian¶

Differentiate $f(x) = e^{-\alpha x^2}$ with respect to $x$.

Problem 2: Product and chain together¶

Differentiate $g(x) = x\sin(kx)$.

Problem 3: A normalization integral¶

Evaluate $\displaystyle\int_0^{L} \sin^2\!\left(\frac{\pi x}{L}\right)dx$.

Problem 4: Integration by parts¶

Evaluate $\displaystyle\int_0^{\infty} x\,e^{-x}\,dx$.

Problem 5: Symmetry shortcut¶

Without computing an antiderivative, evaluate $\displaystyle\int_{-a}^{a} x\,e^{-x^2}\,dx$.

Problem 6: Verify a Riemann sum numerically¶

Modify the Riemann-sum code above to estimate $\int_0^{\pi}\sin x\,dx$ and compare with the exact value 2.

Problem 7: Chain rule on a plane wave¶

Differentiate $\psi(x) = e^{ikx}$ with respect to $x$, and again to find $\psi''(x)$.

Problem 8: An expectation-value integral¶

Evaluate $\displaystyle\int_0^{\infty} x^2\,e^{-x}\,dx$ using integration by parts twice.