Download Quantum Mechanics by Professor Dr. Franz Schwabl (auth.) PDF

By Professor Dr. Franz Schwabl (auth.)

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For example, does the classical energymomentum relation for free particles E = p 2 /2m imply the relation in(ajat) = -(n 2 /2m)V 2 ? 4) for free particles. So, from the classical energy-momentum relation, that is, from the classical Hamiltonian for a free particle, we have obtained the Schrodinger equation for a free particle. This leads us to the quantum mechanical correspondence principle: In quantum mechanics, operators are assigned to physical quantities. Quantum mechanical relationships correspond to classical relationships.

65) This equation is known as the time independent Schrodinger equation. Remarks: The states 1/J(x, t) = exp { -iEt/1i}'lj;(x) are called stationary states, since the corresponding probability densities 11/J(x, t)J2 = l'l/J(x)J2 are time independent. (ii) The condition of normalizability (J d3 xl'l/J(x)J 2 < oo) will limit the allowed values of the energy E. 65) is an eigenvalue equation. We would now like to give a general discussion of these. 66) holds. This equation is called an eigenvalue equation.

5 (a) Show that [A, En]= nBn- 1 [A, B] under the assumption [[A, B], B] = 0. 44). A . 47) holds under the conditions stated in the main text. B . B)- 1 in a power series in),. 8) and verify that the continuity equation is satisfied. 7 Let 1/Ja(x) and 1/Jb(x) be two orthonormal solutions of the time independent Schrodinger equation for a given potential with energy eigenvalues Ea and Eb . At the time t = 0 , suppose that the system is in the state 1/J(x, t = 0) = ~ ( 1/Ja(x) + 1/Jb(x)) Discuss the probability density at a later time t .

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