Principles of Quantum Electronics by Dietrich Marcuse

By Dietrich Marcuse

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We only need to know | 0 > at a particular initial time to know it at all times. The initial value of | 0 > is given by an assumption about the initial state of the 28 Review of Quantum Mechanics CHAPTER ONE system. In many cases one assumes that a precise measurement has been performed so that the state vector initially coincides with one of the eigenvectors. In an alternate mathematical formulation (Heisenberg picture) the state vectors are time-independent and always have the form initially assigned to them.

Both operators correspond to physical variables. 3 measure the variable A we will obtain one of the eigenvalues a of A9 while a measurement of Β must result in an eigenvalue b of B. We assume that the physical system under consideration is described by a state vector \ψ}, which is neither an eigenvector of A nor B. 3-64) can be rewritten with the help of Eq. 3-69) in the same way as Eqs. 3-71) We want to study the uncertainty product AA · Δ £ . Let us instead consider its square which, according to Eqs.

3 A comparison of Eq. 3-54) with Eq. 3-56) The solution of the momentum eigenvalue equation written in the Schrödinger representation can now be obtained from Eqs. , we see that Η and ρ are commuting operators. According to the result of the last section (two commuting operators have the same set of eigenvectors), we must expect that the solution of the momentum eigenvalue problem, Eq. 3-57), is also a solution of the energy eigenvalue equation, Eq. 3-48). The validity of this general result in our special case can easily be checked by substitution of Eq.

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