Questions: The Schrödinger Equation

Question 1

In the time-independent Schrödinger equation Ĥψ = Eψ, what kind of mathematical problem is this?

• A. A differential equation with initial conditions
• B. An eigenvalue equation where E is the eigenvalue and ψ is the eigenfunction
• C. A linear system of algebraic equations
• D. A variational optimization problem

Answer: (B)

Ĥψ = Eψ is exactly the form of an eigenvalue equation: applying the operator Ĥ to ψ returns a scalar multiple E of ψ. The allowed energy levels E are eigenvalues and the corresponding wavefunctions ψ are eigenfunctions of the Hamiltonian operator. This is why linear algebra and eigenvalues are prerequisites.

Question 2

The Schrödinger equation was derived from more fundamental principles, just as Newton's second law can be derived from conservation of momentum.

• T. True
• F. False

Answer: False

The Schrödinger equation is a postulate of quantum mechanics — it was not derived from deeper principles but was proposed by Schrödinger, motivated partly by de Broglie's wave-particle duality. Its justification is empirical: it predicts experimental outcomes with extraordinary accuracy. This is analogous to Newton's second law, which is also a postulate, not a derivation.

Question 3

What is the physical role of the potential energy function V(x) in the time-independent Schrödinger equation, and why does changing it change the allowed energy levels?

Model answer: V(x) describes the force environment the particle inhabits (e.g., a box, a harmonic well, a Coulomb potential). Changing V(x) changes the Hamiltonian operator Ĥ, which changes its eigenvalues, so different potentials yield different discrete energy spectra.

The Hamiltonian is Ĥ = −(ℏ²/2m)d²/dx² + V(x). The kinetic part is fixed for a given mass, but V(x) varies by physical situation. The boundary conditions imposed by V(x) (e.g., ψ = 0 at infinite walls) force the wavefunction to satisfy constraints that only discrete values of E can satisfy — this is the origin of energy quantization.