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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).

is an operator in the sense that it acts on functions of x and t to give functions of x and t: it acts on the space of complex functions, a space that contains wavefunctions. Note that V (x) acts just by multiplication. Note that the operator H. ˆ is time independent – it does not involve time at all. Ψ(x,t) = e ψ −iEt/

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