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) ______ _______ ________ _______ _____ _____ ___ _______ _____ ____ ____ __________.
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___ ________ ____ _____ _________ ____ ________ ___ ________ ____.
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___ ____ ___ ______ _____ ____ _____ __________ ____ ______ ______.
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____ ____ ________ _________ _______ ________ _________ _____ _______ ____.
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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).
A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity , and then its velocity reduces to
, due to the difference in potential. Calculate the transmitted and reflected currents.
If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.
and
are the orthonormal basis states of a two-dimensional Hilbert space. A Hermitian operator
in this basis is given by the spectral representation:
. For a normalized state given by