Analysis and Differential Equations

A differential equation tells us how a function changes. To find a particular solution, we also need information such as its value at an initial time or its behavior at a wall. This chapter develops the calculus used to solve Schrödinger equations, expand wavefunctions, and estimate approximations.

You do not need to finish an analysis course before starting wave mechanics. Begin with ordinary derivatives and integrals, then follow the route that matches your problem.

Enter this chapter

• For a first wave-mechanics course: read Ordinary Differential Equations, Boundary Conditions, and Eigenvalue Problems. You will learn why an equation allows many solutions but a physical problem selects only some of them.
• For wavefunction expansions: start with Sequences, Series, and Convergence and Real Analysis Essentials. These explain what it means for an approximation to approach a state, and when its derivatives or integrals approach the correct answer too.
• For equations in several variables: continue to Partial Differential Equations, Separation of Variables, and Sturm–Liouville Theory. The main idea is to build complicated solutions from simpler spatial modes.
• For sources, approximations, and variational methods: use the final four entries in the page map below when a calculation calls for them.

One equation, different physical problems

Consider

$$-y''(x)=k^2y(x),\qquad 0<x<L.$$

For a given $k>0$, its solutions are $A\cos(kx)+B\sin(kx)$. What happens next depends on the data:

• With $y(0)=1$ and $y'(0)=0$, the solution is $\cos(kx)$ for every $k>0$.
• With $y(0)=y(L)=0$, a nonzero solution exists only when $k=n\pi/L$, with $n=1,2,\ldots$.
• With $y'(0)=y'(L)=0$, cosine modes replace sine modes, and there is also a constant mode at $k=0$.

The first is an initial-value problem; the other two are eigenvalue problems. The equation is the same, but changing the boundary conditions changes the allowed modes. For a particle in a box, multiplying $k^2$ by $\hbar^2/(2m)$ turns these mode values into energies.

Find the method you need

Page	Question it answers
Sequences, Series, and Convergence	What does it mean for an infinite expansion to approach a function or a state?
Real Analysis Essentials	How do continuity, derivatives, and integrability differ?
Complex Analysis Essentials	Why do complex poles and contours help with real physical problems?
Ordinary Differential Equations	How do I solve for a function of one variable?
Partial Differential Equations	What changes when a function depends on space and time?
Boundary Conditions	What must happen at a wall, interface, or distant endpoint?
Eigenvalue Problems	How are allowed energies and modes selected?
Sturm–Liouville Theory	Why are many separated modes orthogonal with a particular weight?
Separation of Variables	When can a PDE be reduced to several ODEs?
Green Functions	How can the response to a point source solve an equation with any source?
Calculus of Variations	How does varying a whole function lead to a differential equation?
Functional Derivatives	How do I calculate the first change in a functional?
Asymptotic Analysis	How can a few terms approximate a function even when the infinite series diverges?

Check your understanding

For $-y''=\lambda y$ with $y(0)=y(L)=0$, why is $y=0$ not called an eigenfunction, and why does normalization not determine $\lambda$?

Solution

The zero function solves the equation for every $\lambda$, so it cannot distinguish any spectral value. An eigenfunction is required to be nonzero. The boundary conditions first select $\lambda_n=(n\pi/L)^2$ and solutions proportional to $\sin(n\pi x/L)$. Normalization then fixes the magnitude of the remaining coefficient; it does not select the energy.

For a first application, try the Infinite Square Well. For approximate solutions on a computer, continue to Numerical Mathematics.

References

• Boyce, William E., Richard C. DiPrima, and Douglas B. Meade. Elementary Differential Equations and Boundary Value Problems. 11th ed. Wiley, 2017. A first introduction to initial-value and boundary-value problems.
• Strauss, Walter A. Partial Differential Equations: An Introduction. 2nd ed. Wiley, 2008. Separation, boundary conditions, and mode expansions.
• Teschl, Gerald. Mathematical Methods in Quantum Mechanics. 2nd ed. American Mathematical Society, 2014. Author’s text. The operator and spectral foundations behind the examples.