CK-12-Physics - Intermediate

24.3. Uncertainty Principle http://www.ck12.org

The uncertainty in the position of the electron is nearly 1,300 times greater than the Bohr radius, which reinforces
the notion of the electron cloud model as opposed to the well-defined planetary atom.

Illustrative Example 25.3.2

The uncertainty in the velocity of a car of mass 1000 kg is 0.75 m/s. What is the minimum uncertainty in the car’s
position?

Solution:

The uncertainty in the momentum of the car is

∆p=m∆v= ( 1000 kg)

(

0. 75

m

s

)

= 750

kg−m

s

Therefore,

∆x≥

h

2 π∆p

=

6. 626 × 10 −^34 J−s

2 π

(

750 kg−sm

) = 1. 41 × 10 −^37 m

Therefore, the minimum uncertainty is∆x= 1. 41 × 10 −^37 m.

Uncertainties this small make it clear why quantum mechanical effects are not seen for macroscopic objects.

The Schrödinger Wave Equation

De Broglie had given a fairly simple way to compute the wavelength and frequency of a matter wave. Computing
the amplitude of the wave, however, was not as simple. In 1925, the physicist Erwin Schrödinger (1887-1961)
succeeded in deriving what is known today as the Schrödinger Wave Equation. The equation provides a way to
calculate the amplitude of an electron wave, and in doing so provides a mechanism that reconciles the particle and
wave aspects of all material objects.

If, using the Schrödinger equation, a particle (say, an electron) is treated as a wave, the equation allows one to find
the amplitude of the wave. If, using the Schrödinger equation, the electron is treated as a particle, the interpretation
of the amplitude of the Schrödinger equation is that the square of the amplitude gives the probability for the electron
being at a specific location in space. These results are very important. Let us restate them:

1. In the first case (electron as wave) the Schrödinger equation gives the amplitude of the matter wave.


2. In the second case (electron as particle) the square of the amplitude assigns a set of probabilities to where the
   electron may be found.

Schrödinger’s equation gave rise to a rather profound question: Are there inherent limits to the precision of mea-
surement?

1. The Balmer series is an empirical formula which gives the reciprocal of the wavelengthλfor each line in the
   hydrogen spectrum
   1
   λ=R

( 1

22 −

1

n^2

)

,n= 3 , 4 ,.. .,

The letterRis known as the Rydberg constant of value

R= 1. 097 × 107 m−^1

The integernis associated with each emission line.

2. Bohr quantized assumptions led to a more general statement of the Balmer series