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Quantum Physics II, Lecture Notes 1 - MIT OpenCourseWare

WAVE MECHANICS B. Zwiebach September 13, 2013 Contents 1 The Schr odinger equation 1 2 Stationary Solutions 4 3 Properties of energy eigenstates in one dimension 10 4 The nature of the spectrum 12 5 Variational Principle 18 6 Position and momentum 22 1 The Schr odinger equation In classical mechanics the motion of a particle is usually described using the time-dependent position ix(t) as the dynamical variable. In wave mechanics the dynamical variable is a wave-function. This wavefunction depends on position and on time and it is a complex number it belongs to the complex numbers C (we denote the real numbers by R).

dimensions of space are relevant we write the wavefunction as . satisfies the Schrodinger equation. For one-dimensional space we write . 2 ∂ 2 . ∂t 2m. ∂x 2 This is the equation for a (non-relativistic) particle of mass m moving along the x axis while acted by the potential V (x, t) ∈ R.

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