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Chapter 3. Wave optics
Experimentally the intensity distribution is a property of the two
sources, i.e., the way in which the system is prepared. The two
sources are said to radiate coherently, because they have a definite, constant phase relationship to one another. This situation
can be realized experimentally by impinging a parallel beam on a
positively charged fiber in otherwise field-free space. The beam is
bent toward the fiber on both sides, thus creating a region where
the beams from the two sides interact coherently with each other.
Such an arrangement is called a biprism, and has been demonstrated. We have seen from this example that a single-particle
state with just two individual eigenstates populated shows striking and distinctive interference behavior.
Problems
1. Write down explicit expressions for the normalized eigenfunctions, eigenvalues, and dispersion relations for a free particle in
one Cartesian dimension, assuming periodic boundary conditions
on a spatial interval of length L.
2. Estimate the quantum number n x for a free electron with energy
1 keV moving in a drift length L = 1 m. The correspondence principle states that quantum mechanical particle motion approaches
classical behavior in the limit of large quantum numbers.
3. The state function Ψ(x, t) is said to describe single-particle
motion in the energy representation, since the eigenvalues of the
ˆ
Hamiltonian operator H represent conserved energy. The state
function Φ(k, t) is said to describe the momentum representation.
Write down explicit expressions for the eigenfunctions and eigenvalues for a free particle in the momentum representation.
Chapter 3. Wave optics
Experimentally the intensity distribution is a property of the two
sources, i.e., the way in which the system is prepared. The two
sources are said to radiate coherently, because they have a definite, constant phase relationship to one another. This situation
can be realized experimentally by impinging a parallel beam on a
positively charged fiber in otherwise field-free space. The beam is
bent toward the fiber on both sides, thus creating a region where
the beams from the two sides interact coherently with each other.
Such an arrangement is called a biprism, and has been demonstrated. We have seen from this example that a single-particle
state with just two individual eigenstates populated shows striking and distinctive interference behavior.
Problems
1. Write down explicit expressions for the normalized eigenfunctions, eigenvalues, and dispersion relations for a free particle in
one Cartesian dimension, assuming periodic boundary conditions
on a spatial interval of length L.
2. Estimate the quantum number n x for a free electron with energy
1 keV moving in a drift length L = 1 m. The correspondence principle states that quantum mechanical particle motion approaches
classical behavior in the limit of large quantum numbers.
3. The state function Ψ(x, t) is said to describe single-particle
motion in the energy representation, since the eigenvalues of the
ˆ
Hamiltonian operator H represent conserved energy. The state
function Φ(k, t) is said to describe the momentum representation.
Write down explicit expressions for the eigenfunctions and eigenvalues for a free particle in the momentum representation.