Workshop Tutorials for Physics
Solutions to QR4: The Uncertainty Principle
A. Qualitative Questions:
1. The Uncertainty principle.
a. px  h/2.
b. The Heisenberg uncertainty principle says that no matter how precise your measurements, the more
you know about one variable, the less it is possible to know about the other, and the product of the two
uncertainties is always greater than or equal to Planck’s constant, h, on 2.
c. All experiments involve some uncertainty due to inaccuracies in measurements, these uncertainties
are also often called “errors”. In a first year physics experiment these uncertainties are enormous
compared to that from the uncertainty relation, so it can be ignored in the first year laboratory.
d. The uncertainty principle places no limit on how accurately you can measure the position or velocity
of an object. It limits how much you can know about position and momentum simultaneously, the more
you know about one, the less you can know about the other.
2. Both Rebecca’s and Brent’s points of view are quite reasonable, given our current knowledge of the
quantum world. Many physicists believe that the uncertainty principle is entirely due to the fact that you
cannot measure something without in some way interacting with it. Einstein said that “God does not play
dice”, meaning that the world is still inherently deterministic.
Many other physicists believe that the universe is not deterministic, and even if you knew
everything about all particles in the universe, you would still not be able to predict the future. Heisenberg
wrote that “In the sharp formulation of the law of causality-- "if we know the present exactly, we can
calculate the future"-it is not the conclusion that is wrong but the premise.” Thus there are at least two
opposing viewpoints on what exactly the uncertainty principle means about the universe, and neither has
as yet been shown to be right.
This has led to a great many philosophical debates on the nature of the universe, free will and the
existence of God. Physicists are still working on the answer!
B. Activity Questions:
1. Measuring momentum and position I
When the marble is released from the top of the slide it rolls down, gathering momentum as it falls. As it
leaves the end of the slide it has horizontal velocity v. It is accelerated vertically due to gravity, and hits
the floor at a time t  ( 2gh )
slide above floor level.
1/ 2
after it leaves the end of the slide, where h is the height of the end of the
1 / 2
The horizontal momentum of the marble during its flight is given by p=mv = mx/t = m x( 2gh )
.
Your measurement has changed the momentum of the marble significantly, in fact it has reduced it to
zero.
You do not know both position and momentum simultaneously; you have found the momentum of the
marble just before it hit the floor. At the time of your measurement you knew its position, but by finding
this you changed the momentum. You know momentum before, and position during the measurement, but
you do not know both simultaneously at the time of the actual measurement. This does not contradict the
uncertainty principle which states that you cannot precisely know both momentum and position
simultaneously.
The Workshop Tutorial Project –Solutions to QR4: The Uncertainty Principle
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2. Measuring momentum and position II
Your measurement has had very little effect on the marble. The scattering of photons from the marble to
the detector will have no measurable effect on the momentum as the change in momentum due to the
measurement is negligible compared to the momentum of the marble.
You do not need to take the uncertainty principle into account in this experiment, as the effects are tiny
compared to the experimental uncertainties involved.
If you were measuring the momentum of an electron using the scattering of light, the momentum transfer
from a photon to the electron, before the photon arrives at the detector, may be significant compared to
the initial momentum of the electron. In this case the uncertainty principle would need to be considered.
C. Quantitative Questions:
1. Life, and cricket, with a large value of Planck’s constant.
a. In both universes there will be an uncertainty due to the accuracy of the equipment Brent is using to
measure the velocity. For example, if he is using a stop-watch to time the ball’s motion he will be limited
by his own reflexes and by the uncertainty in the device, which may only read in seconds or milliseconds.
There will also be an inherent uncertainty due to the wave nature of the ball. The uncertainty principle
tells us that we cannot know both the position and momentum of an object at the same time, px  h/2.
In this universe Planck’s constant is small enough that this will make a negligible difference. However in
the new universe Planck’s constant is very large. There will be a much greater uncertainty in either the
momentum or position (or both) when he attempts to measure the velocity due to the wave nature of the
ball.
b. Assuming no uncertainty in mass, we have p =m v = 1.0 m.s-1  0.5 kg = 0.5 kg.m.s-1,
c. Using px  h/2 we get x  h/(2  0.5 kg.m.s-1)  0.2 m. It would be quite difficult trying to
catch a ball whose position you only know to within 20 cm!
d. The de Broglie wavelength will be  = h/p = 0.6 J.s / 10 kg.m.s-1 = 0.06 m = 6 cm.
You might see the ball diffract from the cricket bat and go around it, or form an interference pattern as it
goes through the wicket, as the wavelength is of similar size to these objects.
2. The x, y and z components of the velocity of an electron are measured to be :
vx = (4.00  0.18)  105 m.s-1 ,
vy = (0.34  0.12)  105 m.s-1 and
vz = (1.41  0.08)  105 m.s-1
a. The momentum is p = mv, assuming that there is no uncertainty in m, then p/p = v/v, which we can
rearrange to give p = p  v/v= mv.
px = mvx = 9.11  10-31 kg × 0.18  105 m.s-1 = 1.64  10-26 kg.m.s-1.
py = mvy = 9.11  10-31 kg  0.12  105 m.s-1 = 1.10  10-26 kg.m.s-1.
and pz = mvz = 9.11  10-31 kg  0.08  105 m.s-1 = 7.29  10-27 kg.m.s-1.
b. Using px  h/2 we can find the smallest uncertainties in the
x, y, and z positions of the electron.
xmin = h/(2  px)
= 6.64  10-34 J.s /(2  1.64  10-26 kg.m.s-1 )
= 6.44  10-9 m = 6.44 nm.
ymin = h/(2  py) = 6.64  10-34 J.s /(2  1.10  10-26 kg.m.s-1) 2ymin
= 9.60  10-9 m= 9.60 nm
2zmin
zmin = h/(2  pz) = 6.64  10-34 J.s /(2 7.29  10-27 kg.m.s-1)
2x
min
= 1.45  10-8 m = 14.5 nm.
The smallest volume to which we can localize the electron is:
V = 2xmin  2ymin  2zmin = 2  6.44  10-9 m  2  9.60  10-9 m  2  1.45  10-8 m = 7.17  10-24 m3
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The Workshop Tutorial Project –Solutions to QR4: The Uncertainty Principle