A quantum mechanical particle in one dimension has the wave function:
Use the Schrödinger equation to determine the corresponding potential V(x).
The time-independent Schrödinger equation in one dimension is [ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi. ] Given (\psi(x) = N x^n) for (x > 0) and (n > 1), we first compute _____ _____ ____ _____ ______ _____ ______ _________ __________ ____ ____.
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i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
The spectral representation of an operator in a two-dimensional orthonormal basis
is
Determine the matrix elements of .
Show that for any stationary state of a symmetric potential well: .
Electrons are accelerated through a potential 150 V and incident on a crystal with interatomic spacing . Calculate the de Broglie wavelength and the first-order Bragg diffraction angle.
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.