# Wave function when there is coupling between spin and position

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• Happiness
So, in summary, the general state, in the presence of coupling, can take the form of a linear combination of the symmetric and anti-symmetric parts of the wave function, with the spin up and spin down spinors representing the two possible orientations of the z-axis. The coefficient of the spin-up spinor does not necessarily need to be symmetric, as it is just a notation for the part of the spatial wave function associated with that spinor.
Happiness

Why can't the general state, in the presence of coupling, take the form $$\psi_-(r)\chi_++\psi_+(r)\chi_-$$ where ##\psi_+(r)## and ##\psi_-(r)## are respectively the symmetric and anti-symmetric part of the wave function, and ##\chi_+## and ##\chi_-## are respectively the spinors representing spin up and spin down? In other words, why must the "coefficient" of the spin-up spinor be symmetric, in the presence of coupling?

Reference: Intro to QM, David J Griffiths, p210

You just changed the orientation of the z axis (by swapping what up and down is). Why would it be symmetric and antisymmetric? I don't know, needs more context, I don't have Griffiths around.

The part of the spatial wave function associated/coupled with ##\chi_+## does not need to be symmetric, right?

On second thought, I think ##\psi_+## does not represent a symmetric wave function, but it’s just to denote the part of the spatial wave function associated with ##\chi_+##. Throughout the book, Griffiths has been using ##\psi_+## to mean a symmetric wave function, but I think it does not mean that in this particular case, hence the confusion.

Last edited:
That makes more sense.

## 1. What is a wave function?

A wave function is a mathematical representation of the quantum state of a particle or system. It describes the probability of finding the particle in a certain position or state and how that probability changes over time.

## 2. How does coupling between spin and position affect the wave function?

Coupling between spin and position means that the spin of a particle is affected by its position and vice versa. This leads to a more complex wave function that takes into account both the position and spin of the particle.

## 3. What is the significance of coupling between spin and position in quantum mechanics?

Coupling between spin and position is a fundamental aspect of quantum mechanics. It helps to explain the behavior of particles on a microscopic level and is essential for understanding phenomena such as electron spin and magnetic properties of materials.

## 4. How is the wave function affected by changes in the coupling between spin and position?

Changes in the coupling between spin and position can alter the shape and behavior of the wave function. This can affect the probability of finding the particle in a certain position or state and can also impact the overall dynamics of the system.

## 5. Can the coupling between spin and position be measured or observed?

Yes, the coupling between spin and position can be measured and observed through various techniques such as electron spin resonance and nuclear magnetic resonance. These techniques allow scientists to study the behavior of particles and systems with coupled spin and position.

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