PHYS 225: Assignment 2

Martin J. Savage

April 11, 2009

Abstract

The second assignment for Introduction to Quantum Mechan-

ics, PHY225, Spring 2009.

April 2009

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Assignment 2: Introduction to Quantum Mechanics, Phys 225due: Monday April 20

Figure 1: The Stern-Gerlach Experiment. Image from Wikipedia

1 Question 1: Pure State

Consider an ensemble of 25 spin-1

2atoms that are all identically prepared in

the (spin) quantum state |ψ1〉,

|ψ1〉 =ei π

3√3|+〉z +

√2 ei π

7√3

|−〉z . (1)

Each atom in the ensemble is passed through a Stern-Gerlach magnet alignedin the +z-direction.

1. Write the bra 〈ψ1| that is dual to the ket |ψ1〉.

2. What is the probability that any given atom emerges from the magnetin the |+〉z state ?

3. What is the probability that any given atom emerges from the magnetin the |−〉z state ?

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If instead of passing through a magnetic oriented in the +z direction, theensemble passed through a magnet oriented in the +y direction.

4. What is the probability that any given atom emerges from the magnetin the |+〉y state ?

5. What is the probability that any given atom emerges from the magnetin the |−〉y state ?

The atoms that emerge from this magnet in the |+〉y state are passed througha second Stern-Gerlach magnet that is oriented in the +z direction.

6. What is the probability that any one of these atoms emerges in the|+〉z state ?

7. Is this result consistent with your answer to part 2 of this question.Explain your answer.

Intrigued by the answers that you have given to the above questions, anexperimentalist is motivated to explore such a system and the set of mea-surements. She constructs the apparatus to have an ensemble of 25 atomsprepared in the |ψ1〉 state pass through the magnet in the +y direction, andthen have those atoms emerging in the |+〉y state pass through a magnetaligned in the +z direction. She observes that ALL 25 of the atoms emergein the |+〉z state !

8. What is the probability of such an observation ?

9. Can you conclude that this experiment is flawed ? Justify your conclu-sion.

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2 Question 2: Mixed State

Consider an ensemble of 25 spin-1

2atoms. Ten (10) of the atoms are identi-

cally prepared in the (spin) quantum state |ψ1〉,

|ψ1〉 =ei π

3√3|+〉z +

√2 ei π

7√3

|−〉z , (2)

while fifteen (15) of the atoms are prepared in the state |ψ2〉

|ψ2〉 = |−〉y . (3)

Each atom in the ensemble is passed through a Stern-Gerlach magnet alignedin the +x-direction.

1. What is the expected number of atoms emerging in the |+〉x state?

2. What are the inner-product 〈ψ2|ψ1〉 and 〈ψ1|ψ2〉 ? Are the states |ψ1〉and |ψ2〉 orthogonal? Are they linearly dependent?

3. Can the states |ψ1〉 and |ψ2〉 be used to construct a spanning-set ofstates for their 2-dimensional-space. If so, construct an orthonormalbasis in terms of |ψ1〉 and |ψ2〉 (and their inner-products), if not explainwhy not.

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