# Stern Gerlach and Position Measurements

• bglasber
In summary, the additional measurement of the particle's position changes its state and therefore affects the probability of detection in D1/D2.
bglasber

## Homework Statement

Given the following stern-gerlach setup, find the probabilities for particles to arrive in detectors D1 and D2.
SGx = Stern Gerlach in x direction,
SGx = Stern Gerlach in z direction
(sorry for terrible ascii drawing)

|ψ> = a|+> + b|->, |ψ> is normalized.
..../D1
.../SGz.\...D2
-- SGx ...SGx/
...\SGz../...\ D3
......\D4

Suppose that you take a measurement of the particle's position after doing the z-measurement so you can tell which of the four paths it took, what is the probability of detection in D1/D2?

## Homework Equations

Chaining projectors, P$_{|+>}$ = |+><+|
Probability: |<ø|ψ>|$^{2}$

## The Attempt at a Solution

Doing the initial probability calculations isn't too bad, we just chain projectors together and calculate the probability for the paths it can take to reach the detectors.
What confuses me though, is the second part, where we place a detector, because I am unsure of how this changes the question.

I thought that we would have already collapsed the wave function after passing through a detector, because the particle would be forced into the measured spin state |+>, |-x>, etc. ( Wouldn't this be the effect of the remaining ket in the projector chain? e.g.
|+> <+|+x> <+x|ψ> for D1?)

I'm guess I don't quite understand how knowing the path that the particle took changes the probability of it getting there? Is my answer supposed to change?

Last edited:

Hi there,

Thank you for your post. It seems like you have a good understanding of the initial probability calculations for the particle's path to reach the detectors. However, you are correct in questioning how the second part of the question, where we place a detector, changes the question.

In this scenario, we are essentially adding another measurement to the system. By placing a detector, we are measuring the particle's position and determining which of the four paths it took. This additional measurement will affect the probability of detection in D1/D2 because it changes the state of the particle.

To understand this better, let's look at the example you provided for D1. In this case, the remaining ket in the projector chain would be |+x>, which represents the particle's state after passing through SGx. However, by placing a detector, we are now measuring the particle's position and determining which of the four paths it took. This means that the particle's state is no longer |+x>, but rather a combination of |+x> and |+z>. This change in state will affect the probability of detection in D1.

I hope this helps clarify things for you. Let me know if you have any further questions.

## What is the Stern Gerlach experiment?

The Stern Gerlach experiment is a physics experiment that was first conducted in 1922 by Otto Stern and Walther Gerlach. It involved passing a beam of silver atoms through a non-uniform magnetic field and observing the deflection of the atoms. This experiment provided evidence for the quantization of atomic angular momentum and was crucial in the development of quantum mechanics.

## How does the Stern Gerlach experiment work?

In the Stern Gerlach experiment, a beam of silver atoms is directed through a non-uniform magnetic field. The atoms are deflected either up or down depending on their orientation of spin. This deflection is due to the magnetic dipole moment of the atoms. The atoms are then collected on a screen or photographic plate, where they form distinct bands or lines, indicating their spin orientation.

## What is the significance of the Stern Gerlach experiment?

The Stern Gerlach experiment provided evidence for the quantization of atomic angular momentum and was crucial in the development of quantum mechanics. It also demonstrated the existence of intrinsic angular momentum, or spin, in particles. This experiment played a crucial role in shaping our understanding of the quantum world and continues to be studied and used in various fields of physics.

## What is a position measurement in the context of the Stern Gerlach experiment?

A position measurement in the Stern Gerlach experiment refers to the act of determining the location of a particle after it has passed through the non-uniform magnetic field. This is typically done by observing the deflection of the particles on a screen or photographic plate. The position of the particles can also be determined by measuring the angle of deflection.

## What are some applications of the Stern Gerlach experiment?

The Stern Gerlach experiment has various applications in fields such as quantum computing, quantum cryptography, and quantum information processing. It is also used in experiments to study the properties of atoms and subatomic particles. Additionally, the principles of the Stern Gerlach experiment have been applied in technologies such as magnetic resonance imaging (MRI) and magnetic data storage.

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