In: Physics
An electron and a proton are both confined in boxes of length a. Measurements are made on the momentum and speed of both particles. Which statement is true about the precision of these measurements
a)The electron’s momentum can be measured more precisely than the proton’s momentum
b)The proton’s momentum can be measured more precisely than the electron’s momentum
c)The electron’s speed can be measured more precisely than the proton’s speed
d)The proton’s speed can be measured more precisely than the electron’s speed.
e)The electron’s speed can measured as precisely as the proton’s speed can be measured.
According to the Uncertainty relation,
And for the uncertainty in the position
And so,
As the right hand side (RHS) does not depend upon any parameter
which might discriminate between electron and proton, so, the
uncertainty in the momentum of the proton and the electron are
equal.
But in the non-relativistic mechanics, we have
And so,
Now as the right hand side (RHS) depends upon the mass of the
particle m, so, the uncertainty in the velocity depends upon the
mass of the particle and so are different for particles of
different masses. So, for the electron, the uncertainty in the
velocity is
And for the proton, the uncertainty in the velocity is
And so, the ratio of the two is
Now as we know
So,
And so, the uncertainty in the velocity of the proton is much less
than the uncertainty in the velocity of the electron.
As the less uncertainty means more preciseness, so, the velocity of
the proton can be measured more precisely than the electron's
velocity.
So, option (d) is correct.