Download Quantum Mechanics by Professor Dr. Franz Schwabl (auth.) PDF
By Professor Dr. Franz Schwabl (auth.)
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Additional info for Quantum Mechanics
Example text
78) m What is the physical significance of the eigenvalues am and the expansion coefficients em? In order to clarify this question, we begin by introducing a few concepts from probability theory. 1 Some Concepts from Probability Theory Let X be an arbitrary random variable 3 taking values x, and w(x)dx the probability that the random variable takes a value in the interval [x, x + dx]. 3 A quantity X is called a random variable if it assumes values x depending on the elements e of a "set of events" E.
Jin Unf3 , which according 34 2. 71) is satisfied. 72) n The 'ljJn thus form a complete set of orthonormal eigenfunctions. 74') n Remark: The Schmidt orthogonaZization procedure for degenerate eigenfunctions 'l/Jl, 'l/J2, .... 70), it is sometimes more convenient to perform a stepwise orthogonalization of a system of degenerate but linearly independent eigenfunctions. Beginning with 'l/Jl, 'l/J2, ... , the functions 'PI, 'P2, ... are defined as follows: 'Pj is constructed from 'l/Jj by projecting out all parts proportional to 'PI, ....
Because the given values are comparable to the dimensions of the effects being investigated, the uncertainties have considerable significance in the atomic domain. 20). Can these also be calculated in coordinate space? X/n]'lj;(X,t) J J (2:n)3'lj;*(X"t)(~V'lj;(X,t)) J p ~(x' p} d3x' eip . x/n'lj;(x,t) d3x'eip . X'/Ii'lj;*(x',t) d3x x d3x' d3 exp { - x) . In the preceding line, we have partially integrated under the assumption that 'lj;(x) falls off sufficiently rapidly at infinity, that is, that the boundary terms are zero.