Consider a Hermitian operator . Show that the operator exp (i alpha ) is unitary

Question

Consider a Hermitian operator . Show that the operator exp (i alpha ) is unitary, where alpha is a real number. Here we define the exponential of an operator as exp . Compute alpha ( ) = exp(-i sigma alpha phi/2), which is a rotation operator by phi around the axis x alpha in the spin-1/2 Hilbert space. sigma alpha, alpha = 1.2. 3, are the Pauli spin matrices Notice that . Verify that alpha( phi ) is unitary and . Find the eigenvectors and eigenvalues of the Pauli spin matrices sigma alpha's, and the rotation operator R alpha (sigma). Find the the unitary operators which diagonalizes the Pauli spin matrix sigma 1 and sigma 2 (see question 3).

Explanation / Answer

A wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves.The most common symbols of a wavefuction are lower case and capital psi.psi^2 is real and corresponds to the probability density of finding a particle in a given place at a given time, if the particle's position is measured.

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