Generalized uncertainty principle

  • #1

Homework Statement


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


Homework Equations



The Attempt at a Solution



I know that the generalized uncertainty principle gives [tex] ΔS_x ΔS_y ≥ |<[S_x, S_y]>|

=> ΔS_x ΔS_y ≥ \frac{h^2}{16π^2} [/tex]
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.
 

Answers and Replies

  • #2
TSny
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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
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
 
  • #4
TSny
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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##?
 
  • #5
How is ##ΔS_x## defined in terms of ##<S^2_x>## and ##<S_x>^2##?
I don't know I'm afraid :/
 
  • #6
TSny
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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:

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