138
Chapter 3. Wave optics
or the distribution of energies. Measurement of these macroscopic
beam properties is conceptually equivalent to repeating the singleparticle experiment many times, once for each particle in the beam.
The preparation of the single-particle state is assumed to be the
same for each particle. This is the conceptual connection between
theory and measurement.
In order to better appreciate the physical significance of the theory, we now explore an important and useful special case, namely,
a particle moving in a field-free space. This is the subject of the
next section.
Problems
1. Construct explicit expressions for the operators representing
the three Cartesian components of angular momentum.
2. By definition, a linear operator C ˆ satisfies
ˆ
ˆ
ˆ
C (c 1 ϕ 1 + c 2 ϕ 2 ) = c 1 C ϕ 1 + c 2 C ϕ 2 ,
(3.49)
where c 1 and c 2 are any two complex constants. Examples of linear
operations include multiplication by a constant and differentiation,
to name just two. Prove that all of the operators discussed in this
section are linear.
C ˆ ˆ
ˆ
3. Prove that the operator C is linear if C is linear.
ˆ
ˆ
4. The commutator of two operators C 1 and C 2 is defined as
[C ˆ 1 , C ˆ 2 ] ≡ C ˆ 1 C ˆ 2 − C ˆ 2 C ˆ 1 .
(3.50)
ˆ
(a) Write down an explicit expression for [ˆ x, P x ], where x ˆ is the
ˆ
operator for the x-coordinate, and P x is the operator for the xcomponent of the canonical momentum.
(b) Write down an explicit expression for [ˆ x, P ˆ y ], where ˆ
x is the
operator for the x-coordinate, and P ˆ y is the operator for the ycomponent of the canonical momentum.
Chapter 3. Wave optics
or the distribution of energies. Measurement of these macroscopic
beam properties is conceptually equivalent to repeating the singleparticle experiment many times, once for each particle in the beam.
The preparation of the single-particle state is assumed to be the
same for each particle. This is the conceptual connection between
theory and measurement.
In order to better appreciate the physical significance of the theory, we now explore an important and useful special case, namely,
a particle moving in a field-free space. This is the subject of the
next section.
Problems
1. Construct explicit expressions for the operators representing
the three Cartesian components of angular momentum.
2. By definition, a linear operator C ˆ satisfies
ˆ
ˆ
ˆ
C (c 1 ϕ 1 + c 2 ϕ 2 ) = c 1 C ϕ 1 + c 2 C ϕ 2 ,
(3.49)
where c 1 and c 2 are any two complex constants. Examples of linear
operations include multiplication by a constant and differentiation,
to name just two. Prove that all of the operators discussed in this
section are linear.
C ˆ ˆ
ˆ
3. Prove that the operator C is linear if C is linear.
ˆ
ˆ
4. The commutator of two operators C 1 and C 2 is defined as
[C ˆ 1 , C ˆ 2 ] ≡ C ˆ 1 C ˆ 2 − C ˆ 2 C ˆ 1 .
(3.50)
ˆ
(a) Write down an explicit expression for [ˆ x, P x ], where x ˆ is the
ˆ
operator for the x-coordinate, and P x is the operator for the xcomponent of the canonical momentum.
(b) Write down an explicit expression for [ˆ x, P ˆ y ], where ˆ
x is the
operator for the x-coordinate, and P ˆ y is the operator for the ycomponent of the canonical momentum.