1. “velocity” is group velocity, not phase velocity
Quantum Waves
Particles are actually waves
Two de Broglie relations
They have all the properties (and equations) that described waves before, together with some new ones
Rewrite the de Broglie relations in terms of k and :
We can also find a dispersion relation for these waves:
From this, we can find the group and phase velocity:
“velocity” is group velocity, not phase velocity

2. Quantum Uncertainty Two uncertainty relations for any type of wave
Multiply by h-bar
Use quantum wave relations
Uncertainty in Position/Momentum
Classical physics:
Particle is defined by position x and velocity v = (dx/dt)
Or, if you prefer, exact x and exact p
Quantum physics:
Particle is defined by wave function (x)
It cannot have a definite position and momentum – they tend to both be uncertain
Uncertainty in Time/Energy
Things that last eternally can have definite energy
Things that last a brief time have uncertain energy
Measurements of energy will have a spread in value

3. Time/Energy Uncertainty
The 0-meson has a rest mass of MeV/c2 and lasts an average of 8.410-17 s. What is the spread in energies of a 0-meson at rest due to its finite lifespan? E E
E - mc2 (eV)

4. These methods give only estimates of the answer
Consequences of Position Uncertainty
Position and momentum of a particle cannot be simultaneously specified or determined
We can often estimate one quantity if we know the other by treating this inequality as if it were an equality
Especially when the energy is being minimized L
Energy of Particle in a Box
By Carlson’s rule, the position is uncertain by about x = ¼L.
By the uncertainty principle, momentum is uncertain by
We’d like the momentum to be zero, but we can’t
This causes the particle to have some energy, called Zero Point Energy ?
These methods give only estimates of the answer

5. Solving Uncertainty/Energy Problems
A particle of mass m lies above an impenetrable barrier in a gravitational field with acceleration g. What is the minimum energy of the particle? y
Write an expression for the energy: potential and kinetic
Figure out what momentum (normally 0) and what position would have the “ideal” lowest kinetic and potential energies
Let x = a and p = /2a
Assume the momentum and position differ from ideal by about x = a and p =/2a
Set derivative of energy function equal to zero, solve for a
Substitute a in to determine minimum energy

6. Uncertainty and the Hydrogen Atom
There are two types of energy associated with the Hydrogen atom
Potential Energy
Kinetic Energy
Classically, these two energies are at a minimum when:
The electron is at rest, p = 0
The electron is at the origin, r = 0 (x = 0)
In this case, the energy of the hydrogen atom E = –
Quantum mechanically, you can’t control both position and momentum
We will place the electron near the origin, r = 0, but there is an uncertainty x associated with it
We will place the electron nearly at rest, p = 0, but there is an uncertainty p associated with it
You have to compromise between these two choices to minimize E
Let x = a, and let p be as small as possible by the uncertainty principle
Plug these into the formula for energy

7. Uncertainty and the Hydrogen Atom
We need to minimize the energy
Take derivative and set it to zero
Substitute result back in
Correct answer is 4 times smaller than this. Why so far off?
It’s just an estimate
There is momentum in three dimensions
This makes answer 3 times smaller
Then answer we get is 4/3 of correct answer

8. “Electric Field is potential for force at a distance on a charge”.
The Wave Function
We’ve been talking about waves, which require wave functions
We’d better give it a name:
It would have more arguments in more dimensions
It is a complex function; it has both a real and imaginary parts
What does it mean?
This turns out to be very hard
Analogy: Electromagnetic Waves
Described by electric and magnetic fields
What is an electric field?
It is defined in terms of a potential for something
American Heritage Science Dictionary
“The distribution in space of the strength and direction of forces that would be exerted on an electric charge at any point in that space”.
Dr. Carlson, PHY 114
“Electric Field is potential for force at a distance on a charge”.

9. The Wave Function: what does it mean?
We are talking about one particle – but it is not at one location in space
If we measured its position, where would we be likely to find it?
The Wave Function is also called the probability amplitude
Clearly, where the wave function is small (or zero), you wouldn’t expect to find the particle
Where it’s negative or imaginary, wouldn’t expect to have negative or imaginary probability
We’d better make darn sure that the probability is always positive
For electric fields, the energy density is proportional to the field squared
If working with complex waves, take amplitude first
How about we make probability density proportional to wave function magnitude squared:

10. Sample Problem
A wave in the region 0 < x < a has the wave function above. What is the probability density at all locations x at all times t?

11. Sample Problem
A wave in the region 0 < x < a has the wave function above. What is the probability density at all locations x at all times t?

12. … in the region 0 < x < a …
What Does Probability Density Mean?
The probability density (in 1D) has units of m-1
In a small region of size dx, the probability of finding the particle is there is given by ||2dx.
To find probability over a larger region, you have to integrate it
Normalization: The probability that the particle is somewhere must be 1
If we integrate over all x, we must get 1
In some cases, the problem implies that we restrict to some region
… in the region 0 < x < a …

13. Sample Problem
At t = 0, the wave function is given by the expression below.
What is the normalization constant N?
What is the probability that the particle is at x > ½a?

14. Sample Problem
At t = 0, the wave function is given by the expression below.
What is the normalization constant N?
What is the probability that the particle is at x > ½a?

15. Sample Problem
At t = 0, the wave function is given by the expression above.
What is the most likely / least likely places to find the particle?
What is the normalization constant N?
What is the probability that the particle is at 0 < x < a?
Least likely when function vanishes, at x = 0
Most likely when function is largest positive or negative
Normalization:
Let x = atan

16. Sample Problem
At t = 0, the wave function is given by the expression above.
What is the most likely / least likely places to find the particle?
What is the normalization constant N?
What is the probability that the particle is at 0 < x < a?

17. End of material for Test 2
Quantum Wave Equations You Need:
End of material for Test 2