Spin 1/2 Particle: Uncertainty Principle Explained

In summary, in Griffiths' book, it is stated that a particle in the ↑ state has a spin angular momentum [Sz] of h bar/2 and the x-component [Sx] can have either h bar/2 or -h bar/2. This is due to the uncertainty principle, which states that both Sz and Sx cannot have a definite value. This can be confusing, as it may seem that measuring one component first affects the other. However, this is because spin is always defined along the z-axis. The particle must have been prepared in the ↑ state with respect to the z-axis.
  • #1
Molar
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In my book (Griffiths) it is said :

A particle is in ↑ state.
The z-component of the particle's spin angular momentum [Sz] is h bar/2.
The x- component of the particle's spin angular momentum [Sx] can be either h bar/2 or
-h bar /2 .

He says this is for uncertainty principle...that both Sz and Sx can't have definite value.
This is where i am having difficulties.

I mean it is always strictly like this that z component would give certain values and uncertainty would follow
x-component...?
or just because we have measured z component first...ie, if we had measured x-component first then we would
have the value h bar /2 for x- component and the uncertainty principle would act on z-component...??
or because spin is always defined along Z axis...?

thanks...
 
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  • #2
Molar said:
or just because we have measured z component first...ie, if we had measured x-component first then we would
have the value h bar /2 for x- component and the uncertainty principle would act on z-component...??
Right.

The particle had to have been prepared in the ↑ state with respect to the z-axis. It could have been any axis--they just like to pick on the z-axis for some reason.
 
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What is a spin 1/2 particle?

A spin 1/2 particle is a fundamental particle that has a spin quantum number of 1/2. This means that it has an intrinsic angular momentum that is equal to half of the Planck's constant. Examples of spin 1/2 particles include electrons, protons, and neutrons.

What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, states that it is impossible to accurately measure both the position and momentum of a particle at the same time. This is because the act of measuring one property will inevitably disturb the other property, leading to an inherent uncertainty in the measurement.

How does the uncertainty principle apply to spin 1/2 particles?

The uncertainty principle applies to all particles, including spin 1/2 particles. In the case of spin 1/2 particles, the uncertainty principle relates to the uncertainty in measuring the spin of the particle along different axes. This means that the more precisely we know the spin along one axis, the less precisely we can know the spin along another axis.

Why is the uncertainty principle important for understanding spin 1/2 particles?

The uncertainty principle is important for understanding spin 1/2 particles because it helps us to understand the fundamental nature of these particles. It also has practical applications in fields such as quantum computing and particle physics, where precise measurements of particle properties are crucial.

Can the uncertainty principle be violated?

No, the uncertainty principle is a fundamental principle of quantum mechanics and cannot be violated. It is a natural consequence of the wave-particle duality of matter and the probabilistic nature of quantum mechanics. However, it is possible to minimize the uncertainty in one property by sacrificing the precision of the other property, but it cannot be completely eliminated.

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