Adding together several plane waves of different wavelengths can produce a wave that is relatively localized.
Note that the uncertainty principle has nothing to do with the precision of an experimental apparatus. Even for perfect measuring devices, these uncertainties would remain because they originate in the wave-like nature of matter. The precise value of the product
depends on the specific form of the wave function. Interestingly, the Gaussian function (or bell-curve distribution) gives the minimum value of the uncertainty product:
The uncertainty principle large and small
Determine the minimum uncertainties in the positions of the following objects if their speeds are known with a precision of
: (a) an electron and (b) a bowling ball of mass 6.0 kg.
Strategy
Given the uncertainty in speed
, we have to first determine the uncertainty in momentum
and then invert
[link] to find the uncertainty in position
.
Solution
For the electron:
For the bowling ball:
Significance
Unlike the position uncertainty for the electron, the position uncertainty for the bowling ball is immeasurably small. Planck’s constant is very small, so the limitations imposed by the uncertainty principle are not noticeable in macroscopic systems such as a bowling ball.
Estimate the ground-state energy of a hydrogen atom using Heisenberg’s uncertainty principle. (
Hint : According to early experiments, the size of a hydrogen atom is approximately 0.1 nm.)
Strategy
An electron bound to a hydrogen atom can be modeled by a particle bound to a one-dimensional box of length
The ground-state wave function of this system is a half wave, like that given in
[link] . This is the largest wavelength that can “fit” in the box, so the wave function corresponds to the lowest energy state. Note that this function is very similar in shape to a Gaussian (bell curve) function. We can take the average energy of a particle described by this function (
E ) as a good estimate of the ground state energy
. This average energy of a particle is related to its average of the momentum squared, which is related to its momentum uncertainty.
Solution
To solve this problem, we must be specific about what is meant by “uncertainty of position” and “uncertainty of momentum.” We identify the uncertainty of position
with the standard deviation of position
, and the uncertainty of momentum
with the standard deviation of momentum
. For the Gaussian function, the uncertainty product is
where
The particle is equally likely to be moving left as moving right, so
. Also, the uncertainty of position is comparable to the size of the box, so
The estimated ground state energy is therefore
