# Generalized uncertainty principle

1. Aug 31, 2013

### DeltaFunction

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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.

2. Aug 31, 2013

### TSny

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 $ΔS_x ΔS_y$?

3. Aug 31, 2013

### DeltaFunction

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

4. Aug 31, 2013

### TSny

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

5. Sep 1, 2013

### DeltaFunction

I don't know I'm afraid :/

6. Sep 1, 2013

### TSny

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.

Last edited: Sep 1, 2013