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Chapter 3. Wave optics
where we can regard Δx as the extent of the wave packet.
In summary, the absolute square |Ψ(x, t)|
2 of the state function
is the probability density that any single measurement will find
a single particle at position x at time t. As such, this quantity
has primary physical significance. The state function, in turn, is
a linear superposition of individual eigenfunctions ψ j (x, t), with
the coefficients a j determined by the way in which the system is
prepared experimentally. The eigenfunctions ψ j with associated
energy eigenvalues ε j represent solutions to the Hamiltonian operator equation, which, in turn governs the dynamical behavior
of the particle. Any single measurement of a single particle must
find the particle in one, and only one eigenstate. In particular, a
single measurement of the particle energy yields one, and only one
eigenvalue H j . The probability of finding the particle in the jth
eigenstate is |a j |
2 .
Problems
1. An electron beam is accelerated to an energy of 1.0 KeV. The
beam is then made to pass through an energy filter which transmits only electrons with a spread of energies ΔE = 0.025 eV
about the mean energy. Estimate the uncertainty in arrival time
of a single electron at a point just at the exit from the energy filter.
2. Electrons are emitted from a cathode and accelerated to form a
beam. Describe in words the conceptual relationship between the
macroscopic beam properties (current, energy, energy spread, path
length, and transit time) and the quantum mechanical motion of
a single beam electron. Assume the beam electrons do not interact
significantly with one another.
Chapter 3. Wave optics
where we can regard Δx as the extent of the wave packet.
In summary, the absolute square |Ψ(x, t)|
2 of the state function
is the probability density that any single measurement will find
a single particle at position x at time t. As such, this quantity
has primary physical significance. The state function, in turn, is
a linear superposition of individual eigenfunctions ψ j (x, t), with
the coefficients a j determined by the way in which the system is
prepared experimentally. The eigenfunctions ψ j with associated
energy eigenvalues ε j represent solutions to the Hamiltonian operator equation, which, in turn governs the dynamical behavior
of the particle. Any single measurement of a single particle must
find the particle in one, and only one eigenstate. In particular, a
single measurement of the particle energy yields one, and only one
eigenvalue H j . The probability of finding the particle in the jth
eigenstate is |a j |
2 .
Problems
1. An electron beam is accelerated to an energy of 1.0 KeV. The
beam is then made to pass through an energy filter which transmits only electrons with a spread of energies ΔE = 0.025 eV
about the mean energy. Estimate the uncertainty in arrival time
of a single electron at a point just at the exit from the energy filter.
2. Electrons are emitted from a cathode and accelerated to form a
beam. Describe in words the conceptual relationship between the
macroscopic beam properties (current, energy, energy spread, path
length, and transit time) and the quantum mechanical motion of
a single beam electron. Assume the beam electrons do not interact
significantly with one another.