Generalized uncertainty principle

[DeltaFunction]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I know that the generalized uncertainty principle gives ΔS_x ΔS_y ≥ |&lt;[S_x, S_y]&gt;|<br /> <br /> =&gt; 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.

 

[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 $ΔS_x ΔS_y$?

 

[DeltaFunction]

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

 

[TSny]

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 $<S_x>^2$?

 

[DeltaFunction]

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

I don't know I'm afraid :/

 

[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.