# Generalized uncertainty principle

• DeltaFunction
In summary, the problem is asking to show that the product ##<S^2_x> < S^2_y>## is consistent with the uncertainty principle, which states that ##ΔS_x ΔS_y ≥ \frac{h^2}{16π^2}##. This can be demonstrated by evaluating ##(\langle S_x^2\rangle-\langle S_x\rangle^2)^{1/2}## for a state of definite z-component of spin.

## Homework Statement

A spin-half particle is in a known eigenstate of Sz. Show that the product $$<S^2_x> < S^2_y>$$ is consistent with the Uncertainty principle

## The Attempt at a Solution

I know that the generalized uncertainty principle gives $$ΔS_x ΔS_y ≥ |<[S_x, S_y]>| => ΔS_x ΔS_y ≥ \frac{h^2}{16π^2}$$
but I'm stuck as to how to proceed to show the product in question is consistent with this.

Thanks in advance for any help.

Hello, DeltaFunction.

I'm not quite sure what you are asking. Are you asking how to evaluate the product ##<S^2_x> < S^2_y> ##, or are you asking how the product ##<S^2_x> < S^2_y> ## is related to ##ΔS_x ΔS_y##?

TSny said:
Hello, DeltaFunction.

I'm not quite sure what you are asking. Are you asking how to evaluate the product ##<S^2_x> < S^2_y> ##, or are you asking how the product ##<S^2_x> < S^2_y> ## is related to ##ΔS_x ΔS_y##?

I'm a bit vague on this too. The problem simply states "Show that the product ##<S^2_x> < S^2_y> ## is consistent with the uncertainty principle"

I assumed I'd need to link that to ##ΔS_x ΔS_y## to demonstrate this, but I'm stumped by this

DeltaFunction said:
I'm a bit vague on this too. The problem simply states "Show that the product ##<S^2_x> < S^2_y> ## is consistent with the uncertainty principle"

I assumed I'd need to link that to ##ΔS_x ΔS_y## to demonstrate this, but I'm stumped by this

How is ##ΔS_x## defined in terms of ##<S^2_x>## and ##<S_x>^2##?

TSny said:
How is ##ΔS_x## defined in terms of ##<S^2_x>## and ##<S_x>^2##?

I don't know I'm afraid :/

You need to know what quantitative definition of "uncertainty" you are using. Generally, the uncertainty of an observable ##A## for a particular state is defined to be the standard deviation $$\Delta A=(\langle A^2\rangle-\langle A\rangle^2)^{1/2}$$
So to determine the uncertainty ##\Delta S_x## for a state of definite z-component of spin, you will need to evaluate ##(\langle S_x^2\rangle-\langle S_x\rangle^2)^{1/2}## for that state.

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