Download ANGULAR MOMENTUM IN QUANTUM MECHANICS

ANGULAR MOMENTUM IN QUANTUM MECHANICS
I.
Orbital angular momentum
A. Consider a particle in a three-dimensional potential that has quantum number l = 3.
1. Suppose you were to measure the magnitude of the orbital angular momentum vector for
this particle. What value(s) could you get? Explain.
2. Can the magnitude of the orbital angular momentum vector for a particle be negative?
Can the quantum number l for a particle be negative?
The orbital angular momentum vector satisfies the following eigenvalue equation:
L̂2 l, ml = l(l +1) 2 l, ml where l is a nonnegative integer.
3. Resolve any inconsistencies between this expression and your answers above.
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B. Consider a classical vector A with magnitude 3.5. The direction of A is not known.
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1. Determine the largest possible value for the z-component of A .
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2. Determine the smallest possible value for the z-component of A . (Hint: Can a single
component of a vector be negative?)
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3. Determine the set of all possible values for the z-component of A . Explain.
Tutorials in Physics: Quantum Mechanics
©McDermott, Heron, Shaffer, and P.E.G., U. Wash.
Preliminary First Edition, 2014
QM
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QM Angular momentum in quantum mechanics
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C. Suppose you were to measure the magnitude of the z-component of the orbital angular
momentum vector for a particle that has l = 3. Determine the set of all possible value(s) for
this measurement. Explain. (Hint: Use your argument from part B, but recall the quantization
condition for angular momentum in quantum mechanics.)
The z-component of the orbital angular momentum satisfies the following eigenvalue equation:
Lˆ z l , ml
z
= ml ! l , ml
z
where ml is an integer with ml ≤ l.
D. Determine the set of all possible values that could result from a measurement of the
x-component of the particle’s angular momentum. Assume no other measurements have been
made. Explain. (Hint: Is there anything unique about the z-direction?)
Would your answer change if you were to consider a measurement of the component of
angular momentum along an arbitrary axis (e.g., the y-axis, or halfway between the x- and
z-axes)?
E. Do you agree or disagree with the following statement? Explain.
”When we measure the z-component of L, we always get something less than the total L, so
angular momentum can never point in the z-direction. But if we happened to choose the direction
in which L did point, then we could measure the full magnitude.”
F. Consider a particle for which only the quantum number ml is known. What are the possible
values of the quantum number l for this particle? Explain.
! Check your results with a tutorial instructor.
Tutorials in Physics: Quantum Mechanics
©McDermott, Heron, Shaffer, and P.E.G., U. Wash.
Preliminary First Edition, 2014
Angular momentum in quantum mechanics QM
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III. Angular momentum and uncertainty
Consider a particle with the initial orbital angular momentum state ψ = l, ml
z
= 2,1 z .
A. Determine two quantities related to orbital angular momentum that are well-defined for this
particle, and the value for each. Explain.
B. Is the x-component of the orbital angular momentum, Lx, well-defined for the initial state
given above? Explain.
If Lx is well-defined, what is its value? If it is not well-defined, determine the possible results
of a measurement of the x-component of the orbital angular momentum. Explain.
What additional information would you need to determine the probabilities of the possible
results of this measurement? Explain.
C. Suppose instead that you were to measure the y-component of the orbital angular momentum,
Ly. Would your answers to part B change? Explain.
Suppose that a measurement of the y-component of the orbital angular momentum of the initial
state above resulted in +ħ.
D. Consider the student discussion below.
Student 1: ”We started knowing the z-direction was 1ħ, and now we know the y-direction is also 1ħ.
The magnitude of angular momentum is only 2ħ, so the x-component must be 0.”
Student 2: ”Angular momentum squared is l(l +1), which for this state is 6ħ2. I know that
L2 = L2x + L2y + L2z ,
so if y and z are both 1ħ, I get that the x-component must be 2ħ.”
Neither statement is completely correct. Explain where each statement is incorrect.
Tutorials in Physics: Quantum Mechanics
©McDermott, Heron, Shaffer, and P.E.G., U. Wash.
Preliminary First Edition, 2014
QM Angular momentum in quantum mechanics
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Suppose that a measurement in the x-direction resulted in -2ħ; e.g., ψ = l, ml
x
= 2, −2 x .
E. Suppose you made a second measurement, this time of the z-component.
1. Predict the ranking of the probabilities of all the possible results of this measurement
from most probable to least probable. Explain your reasoning.
The angular momentum state of this particle can be rewritten in terms of the z-basis, as
follows: 2, −2
x
=
1
2, 2 z − 2 2,1 z + 6 2, 0 z − 2 2, −1 z + 2, −2
4
(
z
).
2. Use the expression above to determine the probability associated with each value of Lz for
this particle. Show your work. Resolve any inconsistencies with your prediction.
All three components of a quantum mechanical angular momentum vector cannot be welldefined at any given instant; however, classical reasoning can still help make predictions about
the probabilities associated with the allowed values of an angular momentum measurement.
F. Use your knowledge of classical vectors to account for each of the following pieces of
information about the particle above:
1. The most likely result of a measurement of Lz is 0.
2. The probabilities are the same for positive and negative results with the same absolute
value.
3. The probabilities are not equally distributed among the possible values.
According to the uncertainty principle, only one of the components of angular momentum in
quantum mechanics may be known precisely at any one instant. If a particle is in an eigenstate of
L̂z then a measurement along a perpendicular axis, such as L̂x , can (usually) result in any
allowed eigenvalue of L̂x . You will investigate this further in the homework.
Tutorials in Physics: Quantum Mechanics
©McDermott, Heron, Shaffer, and P.E.G., U. Wash.
Preliminary First Edition, 2014