Transcription of Quantum Physics II, Lecture Notes 1 - MIT OpenCourseWare
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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). When all three dimensions of space are relevant we write the wavefunction as (ix, t) C . ( ) When only one spatial dimension is relevant we write it as (x, t) C.
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. It is clear from this equation that the wavefunction must be complex: if it were real, the right-hand side of (1.2) would be real while the left-hand side would be imaginary, due to the explicit factor of i. ...
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