i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
The normalization condition requires (\int_0^L |\psi(x,0)|^2,dx = 1). Given (\psi(x,0)=N\left[\sin\left(\frac{\pi x}{L}\right)+\sin\left(\frac{2\pi x}{L}\right)\right]), [ \int_0^L |\psi|^2 dx = N^2 \int_0^L \left[\sin^2!\left(\frac{\pi x}{L}\right)+\sin^2!\left(\frac{2\pi x}{L}\right) +2\sin!\left(\frac{\pi x}{L}\right)\sin!\left(\frac{2\pi x}{L}\right)\right] dx. ] ____ __________ ___ ___ __________ _______ _______ _____ ____ ________.
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The spectral representation of an operator in a two-dimensional orthonormal basis
is
Determine the matrix elements of .
For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
Show that in the terms of the and
basis vectors defined by
;
, we can write:
i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
A quantum mechanical particle in one dimension has the wave function:
Use the Schrödinger equation to determine the corresponding potential V(x).