# The Uncertainty Principle

The Uncertainty Principle states that it is impossible to find out the position and the momentum of an electron at the same instant in time.

### Introduction

The Heisenberg uncertainty principle says that in quantum mechanics, some pairs of physical properties such as position and momentum cannot be measured together in great accuracy. If one of the properties is measure in high accuracy, the other will be measured in less accuracy.

### General Uncertainty Principle

Let's start off with this equation: ∆x∆p ≥ ħ/2

delta x is the uncertainty in the position

delta p is the uncertainty in linear momentum

h with the bar over it is the reduced planck's constant, with the value of 1.052x10^-34 J*s

As you can see, the uncertainty in the position is inversely proportional to the uncertainty in momentum. This means that as you become more accurate about either x or p, you will become more uncertain about the other variable.

### Uncertainty Principle with Quantum Mechanics

The allowed energy levels for a particle in a box is: En = (n2h2)/(8mL2)

You can derive this equation using the Schrödinger wave equation and few other general mathematical equations.

n=1,2,3,... and this is the quantum number, which labels the state of the system.

h is the Planck's constant. Notice that this is not a reduced form from the above section. h=6.626x10^-34 J*s

m is the mass of the particle

L is the length of the one dimensional box

Here are the wave functions

of a particle in a box with n=1,2,3:

when n=1, E1=h2/8mL2    There is no node present

n=2, E2=4h2/8mL2    There is 1 node present

n=3, E3=9h2/8mL2   There are 2 nodes present

### References

1. Housecroft, Catherine E. and Alan G. Sharpe, “Inorganic Chemistry,” 3rd ed. England: Pearson Education Limited: 2008.
2. Ralph H. Petrucci, William S. Harwood, Geoff E. Herring, Jeffry Madura, "General Chemistry," 9th ed. Prentice Hall: 2007

• en.wikipedia.org/wiki/Particle_in_a_box
• en.wikipedia.org/wiki/Uncertainty_principle

### Problems

Question: Estimate the minimum uncertainty in the position of each of an electron in a hydrogen item, given that their speeds are known to within 1.0 μm/s

Question: Calculate the energies of the two states of lowest energy for an electron in a one dimensional box of length 2.0 Å.
Answer: E1=1.506x10^18 J, E2=6.024 x10^18 J

Question: The positions of two objects are measured to the same accuracy while their velocities are measured as accurately as the uncertainty principle allows. If one object has ten times the mass   of the other, the more massive object has a velocity uncertainty which is _(answer)_ the velocity uncertainty of the less massive object.
(A) 1/100
(B) 1/10
(C) the same as
(D) 10 times

### Contributors

• Name #1 here (if anonymous, you can avoid this) with university affiliation

10:17, 2 Oct 2013

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