i) Write down the eigenfunctions for the and
states of the hydrogen atom.
ii) Write the eigenvalues of and
for the states
and
.
iii) Calculate the most probable value of r for the hydrogen atom in the state .
For the hydrogen atom the stationary states are written as ψₙₗₘ(r,θ,φ)=Rₙₗ(r)Yₗₘ(θ,φ). i) For ψ₃₀₀ we have n=3, l=0, m=0. The angular part is the spherical harmonic Y₀₀(θ,φ)=1/√(4π). The radial part for __________ ___ ____ ____ _______ __________ ________ ___.
__________ ______ ______ ___ __________.
________ ___ __________ _______ ______ _____ ____.
______ _______ _________ _________ __________ __________ ___ ___ ___ ____ ______ ____.
________ _______ ____ ________ _________ __________ _______ ____ _____ _____ __________ ____.
______ ______ __________ ________ ___ _________ ______.
_______ _____ _________ _____ ______ _______ ____ _______ ________ ____ ___ ___.
____ _________ _________ _________ ______ ____ ___ _________ _________.
___ __________ ______ _____ ____.
_________ _____ _________ _____ ________ _________ ________ ___ ___ ____ ________.
___ ____ ____ ________ __________ ___ _________ ________ ___ ________.
_______ ________ _________ __________ ____.
_________ _____ __________ ________ ___ ____ ________ _________.
___ _______ ___ _________ ___ ____ _____ ______ _______ ________.
_______ ____ ________ ________ __________ __________ _______ ________.
___ _____ ______ ___ _________.
______ _________ ____ ____ _______ _______ ________ __________ __________.
_____ _______ ______ _______ __________ _____ ___ _________.
__________ _______ _______ ___ ____ ______.
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For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element
i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.
Estimate the uncertainty in the energy of a photon localized within a distance of .
The spectral representation of an operator in a two-dimensional orthonormal basis
is
Determine the matrix elements of .