Uncertainty Calculation for Quantum Physics Lab | General Physics FAQ

[Zarlucicil]

I'm in the process of doing a quantum physics lab and am having a bit of trouble with uncertainty. The specific things going on in the lab aren't relevant, I don't think, only the general procedure of my calculation. Also, I'm not certain where this question should be asked, so I decided to put it here (General Physics).

Let's say we are measuring a quantity x that has an uncertainty of b. Thus,
$$x \pm b$$.

Clearly, x has a maximum and minimum value. Namely,
$$x_L = x - b$$
$$x_H = x + b$$.

There is another quantity y, which is constant, that has an uncertainty of d. It appears y also has a maximum and minimum value,
$$y_L = y - d$$
$$y_H = y + d$$.

Now we want to multiply these two quantities together to get a new quantity z = xy. It seems that this value z has a maximum and minimum value as well,
$$z_L = y_L x_L$$
$$z_H = y_H x_H$$.

Is it permissible to say that the uncertainty on z is the following? (The length between the two extreme values of z divided by 2)
$$\frac{z_H - z_L}{2}$$.

So, after we measure x we need to multiply by y to get z, and y is a constant which only needs to be measured once, but still as an uncertainty. Can we describe z in the following manner?

$$z \pm \left( \frac{z_H - z_L}{2} \right)$$

 

[bp_psy]

What I would use is the partial derivatives method. What you have is f(x,y)=z=xy then $$\sigma_{z}^{2}=\sigma_{x}^{2} (\frac{\partial z}{\partial x})_{x,y}^2+ \sigma_{y}^{2} (\frac{\partial z}{\partial y})_{x,y}^2$$ will give in your case: $$\sigma_{z}=\sqrt{d^{2}x^2 +b^2 y^2}$$.This is good only if your x and y do not have covariance.

 

[jtbell]

Actually, you don't need partial derivatives for this. See posts #6 and #8 in the following thread:

https://www.physicsforums.com/showthread.php?t=459048