Calculus and Differential Equations Checklist

Calculus and differential equations enter quantum mechanics whenever states are represented by wavefunctions. Normalization uses integrals, observables often use differential operators, and energy levels often come from boundary-value problems.

This checklist focuses on the skills needed to read wave-mechanics and canonical-system pages without turning each page into a calculus review.

You Should Be Able To

• Differentiate and integrate elementary real and complex-valued functions.
• Use integration by parts.
• Normalize functions on an interval or on the real line.
• Distinguish initial-value problems from boundary-value problems.
• Solve simple first-order and second-order ordinary differential equations.
• Apply boundary conditions such as fixed endpoints, periodicity, or square integrability.
• Recognize eigenvalue problems for differential operators.
• Use separation of variables in simple partial differential equations.
• Non-dimensionalize an equation using natural length and energy scales.
• Check units and limiting cases in a differential equation.

Why This Matters in Quantum Mechanics

The time-dependent Schrödinger equation is an evolution equation. The time-independent Schrödinger equation is usually an eigenvalue equation. Boundary conditions select the allowed domain of the Hamiltonian and can quantize energy. Normalization and expectation values require integrals. Probability current and continuity equations require derivatives.

The most common mistake is treating a differential expression as the whole problem. In wave mechanics, the equation, interval, domain, and boundary conditions all matter.

Minimum Examples

You should be comfortable with:

Task	Example
Normalize a function	$\psi(x)=A\sin(\pi x/L)$ on $0<x<L$
Solve a boundary-value problem	$\psi''+k^2\psi=0$ with $\psi(0)=\psi(L)=0$
Recognize an eigenvalue equation	$H\psi=E\psi$
Use integration by parts	prove a simple differential operator is symmetric under boundary assumptions
Separate variables	$\Psi(x,t)=\psi(x)T(t)$
Scale variables	introduce $\xi=x/a$ for a natural length $a$

Diagnostic Problems

1. Normalize $\psi(x)=A\sin(\pi x/L)$ on $0<x<L$.

Solution

Normalization requires

$$1=\int_0^L \lvert A\rvert^2\sin^2(\pi x/L)\,dx =\lvert A\rvert^2\frac{L}{2}.$$

Taking $A$ real and positive gives $A=\sqrt{2/L}$.

2. Solve $\psi''+k^2\psi=0$ with $\psi(0)=\psi(L)=0$ and nonzero $\psi$.

Solution

The general solution is $\psi(x)=A\sin(kx)+B\cos(kx)$. The condition $\psi(0)=0$ gives $B=0$. The condition $\psi(L)=0$ gives $\sin(kL)=0$, so $kL=n\pi$ for positive integer $n$. Thus $k=n\pi/L$.

3. For a one-dimensional Hamiltonian

$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$$

why is it not enough to write only this expression?

Solution

The differential expression does not specify the operator domain. The interval, boundary conditions, regularity assumptions, and behavior at singular points or infinity affect which wavefunctions are allowed and which spectra occur.

4. If $x=a\xi$ with dimensionless $\xi$, how does $d^2/dx^2$ transform?

Solution

Since $d/dx=(1/a)d/d\xi$, the second derivative transforms as

$$\frac{d^2}{dx^2} =\frac{1}{a^2}\frac{d^2}{d\xi^2}.$$

This is why choosing a natural length scale also fixes a natural kinetic-energy scale.

Where to Review

Use these pages when a checklist item is weak:

• Real Analysis Essentials
• Ordinary Differential Equations
• Partial Differential Equations
• Boundary Conditions
• Eigenvalue Problems
• Separation of Variables
• Sturm-Liouville Theory
• Time-Independent Schrödinger Equation

References

• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• E. Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley, 2011.