How Does Optical Radiation Affect Electron Velocity Uncertainty?

[clambake]

Homework Statement

Suppose optical radiation (gamma = 5.00 x 10^-7 m) is used to determine the position of an electron to within the wavelength of the light. What will be the resulting uncertainty in the electron's velocity?

wavelength = 5.00 x 10^ -7 m
mass of electron = 9.11 x 10^ -31 kg

Homework Equations

gamma = h/p = h/mv
(delta x)(delta p) = h/4pi

The Attempt at a Solution

I solved for momentum and got 1.33 x 10^-27 kg. I tried to then plug that value back into the equation and solve for velocity, but the answer is way off. I am sure I need to use the uncertainty principle equation, but I'm not sure where to go. I can solve for delta x, and then...?

 

[alphysicist]

Hi clambake,

You do need to use the uncertaintly principle equation. They want the uncertainty in the electron's velocity, so you need to use the equation to find the uncertainty in the momentum (delta p).

To get that, you first need the uncertainty in the position (delta x). That (delta x) will depend upon how the measurement is being made. What does the problem statement say about the measurement uncertainty?

 

[Moetasim]

The Uncertainty Principle states that it is impossible to know the exact position and momentum of a particle simultaneously. This means that when we try to measure the position of an electron using optical radiation, there will always be a certain level of uncertainty in its momentum.

In this case, the uncertainty in position is equal to the wavelength of the light used (5.00 x 10^-7 m). Using the Uncertainty Principle equation, we can solve for the uncertainty in momentum:

(delta x)(delta p) = h/4pi

(5.00 x 10^-7 m)(delta p) = h/4pi

delta p = h/(4pi*5.00 x 10^-7 m) = 1.05 x 10^-24 kg m/s

Now, we can use this value of uncertainty in momentum to calculate the uncertainty in velocity:

(delta x)(delta p) = h/4pi

(delta x)(m*delta v) = h/4pi

(5.00 x 10^-7 m)(9.11 x 10^-31 kg)(delta v) = h/4pi

delta v = h/(4pi*5.00 x 10^-7 m*9.11 x 10^-31 kg) = 3.68 x 10^3 m/s

Therefore, the resulting uncertainty in the electron's velocity is 3.68 x 10^3 m/s. This means that even though we can determine the position of the electron to within the wavelength of the light, we cannot know its exact velocity and there will always be a level of uncertainty.