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. The wavefunction satisfies the Schr odinger equation.
This probability density so defined is positive. The physical interpretation of the wavefunction arises because we declare that . P (x, t)dx is the probability to find the particle in the interval [xx,+ dx] at time t. ... in our present one-dimensional case, it has ... (but are piece-wise continuous, like the finite square well)
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