The Schrodinger Equation

The Equation

The time-dependent Schrodinger equation is:

i\hbar \frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)

where:

• \Psi(x,t) is the wave function
• \hat{H} is the Hamiltonian operator
• \hbar is the reduced Planck constant

Mathematical Structure

Hilbert Space Framework

The wave function \Psi lives in a Hilbert space \mathcal{H} = L^2(\mathbb{R}^3), the space of square-integrable functions:

\langle \Psi | \Psi \rangle = \int |\Psi(x)|^2 \, d^3x < \infty

The Hamiltonian

For a non-relativistic particle in a potential V(x):

\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(x)

This is a self-adjoint operator on \mathcal{H}.

Time Evolution

The formal solution is:

\Psi(t) = e^{-i\hat{H}t/\hbar}\Psi(0)

The operator U(t) = e^{-i\hat{H}t/\hbar} is unitary, preserving probability.

Time-Independent Case

For stationary states with definite energy E:

\hat{H}\psi = E\psi

This is an eigenvalue problem. The full solution becomes:

\Psi(x,t) = \psi(x)e^{-iEt/\hbar}

Connection to Classical Mechanics

Classical	Quantum
Position x	Operator \hat{x}
Momentum p	Operator \hat{p} = -i\hbar\nabla
Hamiltonian H(x,p)	Operator \hat{H}(\hat{x},\hat{p})
Poisson bracket	Commutator \frac{1}{i\hbar}[\cdot,\cdot]

Limitations

The Schrodinger equation is:

• Non-relativistic: Not compatible with special relativity
• Single-particle: Multi-particle versions exist but become unwieldy

These limitations motivate Quantum Field Theory, where particles are excitations of underlying fields.

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References

• E. Schrödinger, “Quantisierung als Eigenwertproblem,” Annalen der Physik 79, 361–376 (1926) — Original paper
• D. J. Griffiths, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, 2018), Chapters 1–2
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed. (Addison-Wesley, 2011)
• R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, 1994)
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vols. 1–2 (Wiley, 1977)