Calculate the matrix elements for J2 for a system of two spin half particles.
For a system of two spin-(\frac{1}{2}) particles, the total angular momentum operator is [ \vec{J} = \vec{S}_1 + \vec{S}_2 ] and the square of the total angular momentum is [ J^2 = (\vec{S}_1 + \vec{S}_2)^2 = S_1^2 + S_2^2 + 2 \vec{S}_1 \cdot \vec{S}_2. ] Each spin-(\frac{1}{2}) particle has [ S_1^2 = S_2^2 = s(s+1)\hbar^2 = \frac{3}{4}\hbar^2. ] The basis for two spin-(\frac{1}{2}) particles can be written as the product states: __________ _______ _____ ___ ____ ____ ____ _______ ____ _________ _____.
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A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with . At t > 0, an electric field
acts on the particle where
is a constant. If the particle is in the ground state when t < 0, calculate the probability that it will be in the first excited state for t > 0.
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and
. Show that
and determine the eigenvalues of
.
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with
.
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.