1. Chapter 41
Quantum Mechanics

2. Quantum Mechanics
The theory of quantum mechanics was developed in the 1920s
By Erwin Schrödinger, Werner Heisenberg and others
Enables use to understand various phenomena involving
Atoms, molecules, nuclei and solids

3. Probability – A Particle Interpretation
From the particle point of view, the probability per unit volume of finding a photon in a given region of space at an instant of time is proportional to the number N of photons per unit volume at that time and to the intensity

4. Probability – A Wave Interpretation
From the point of view of a wave, the intensity of electromagnetic radiation is proportional to the square of the electric field amplitude, E
Combining the points of view gives

5. Probability – Interpretation Summary
For electromagnetic radiation, the probability per unit volume of finding a particle associated with this radiation is proportional to the square of the amplitude of the associated em wave
The particle is the photon
The amplitude of the wave associated with the particle is called the probability amplitude or the wave function
The symbol is ψ

6. Wave Function
The complete wave function ψ for a system depends on the positions of all the particles in the system and on time
The function can be written as
rj is the position of the jth particle in the system
ω = 2πƒ is the angular frequency

7. Wave Function, cont. The wave function is often complex-valued
The absolute square |ψ|2 = ψ*ψ is always real and positive
ψ* is the complete conjugate of ψ
It is proportional to the probability per unit volume of finding a particle at a given point at some instant
The wave function contains within it all the information that can be known about the particle

8. Wave Function Interpretation – Single Particle
Y cannot be measured
|Y|2 is real and can be measured
|Y|2 is also called the probability density
The relative probability per unit volume that the particle will be found at any given point in the volume
If dV is a small volume element surrounding some point, the probability of finding the particle in that volume element is
P(x, y, z) dV = |Y |2 dV

9. Wave Function, General Comments, Final
The probabilistic interpretation of the wave function was first suggested by Max Born
Erwin Schrödinger proposed a wave equation that describes the manner in which the wave function changes in space and time
This Schrödinger wave equation represents a key element in quantum mechanics

10. Wave Function of a Free Particle
The wave function of a free particle moving along the x-axis can be written as ψ(x) = Aeikx
A is the constant amplitude
k = 2π/λ is the angular wave number of the wave representing the particle
Although the wave function is often associated with the particle, it is more properly determined by the particle and its interaction with its environment
Think of the system wave function instead of the particle wave function

11. Wave Function of a Free Particle, cont.
In general, the probability of finding the particle in a volume dV is |ψ|2 dV
With one-dimensional analysis, this becomes |ψ|2 dx
The probability of finding the particle in the arbitrary interval a £ x £ b is
and is the area under the curve

12. Wave Function of a Free Particle, Final
Because the particle must be somewhere along the x axis, the sum of all the probabilities over all values of x must be 1
Any wave function satisfying this equation is said to be normalized
Normalization is simply a statement that the particle exists at some point in space

13. Expectation Values
Measurable quantities of a particle can be derived from ψ
Remember, ψ is not a measurable quantity
Once the wave function is known, it is possible to calculate the average position you would expect to find the particle after many measurements
The average position is called the expectation value of x and is defined as

14. Expectation Values, cont.
The expectation value of any function of x can also be found
The expectation values are analogous to weighted averages

15. Summary of Mathematical Features of a Wave Function
ψ(x) may be a complex function or a real function, depending on the system
ψ(x) must be defined at all points in space and be single-valued
ψ(x) must be normalized
ψ(x) must be continuous in space
There must be no discontinuous jumps in the value of the wave function at any point

16. Particle in a Box
A particle is confined to a one-dimensional region of space
The “box” is one- dimensional
The particle is bouncing elastically back and forth between two impenetrable walls separated by L
Please replace with fig a

17. Potential Energy for a Particle in a Box
As long as the particle is inside the box, the potential energy does not depend on its location
We can choose this energy value to be zero
The energy is infinitely large if the particle is outside the box
This ensures that the wave function is zero outside the box

18. Wave Function for the Particle in a Box
Since the walls are impenetrable, there is zero probability of finding the particle outside the box
ψ(x) = 0 for x < 0 and x > L
The wave function must also be 0 at the walls
The function must be continuous
ψ(0) = 0 and ψ(L) = 0

19. Wave Function of a Particle in a Box – Mathematical
The wave function can be expressed as a real, sinusoidal function
Applying the boundary conditions and using the de Broglie wavelength

20. Graphical Representations for a Particle in a Box

21. Active Figure 41.4
Use the active figure to measure the probability of a particle being between two points for three quantum states
PLAY
ACTIVE FIGURE

22. Wave Function of the Particle in a Box, cont.
Only certain wavelengths for the particle are allowed
|ψ|2 is zero at the boundaries
|ψ|2 is zero at other locations as well, depending on the values of n
The number of zero points increases by one each time the quantum number increases by one

23. Momentum of the Particle in a Box
Remember the wavelengths are restricted to specific values
l = 2 L / n
Therefore, the momentum values are also restricted

24. Energy of a Particle in a Box
We chose the potential energy of the particle to be zero inside the box
Therefore, the energy of the particle is just its kinetic energy
The energy of the particle is quantized

25. Energy Level Diagram – Particle in a Box
The lowest allowed energy corresponds to the ground state
En = n2E1 are called excited states
E = 0 is not an allowed state
The particle can never be at rest

26. Active Figure 41.5 Use the active figure to vary
The length of the box
The mass of the particle
Observe the effects on the energy level diagram
PLAY
ACTIVE FIGURE

27. Boundary Conditions
Boundary conditions are applied to determine the allowed states of the system
In the model of a particle under boundary conditions, an interaction of a particle with its environment represents one or more boundary conditions and, if the interaction restricts the particle to a finite region of space, results in quantization of the energy of the system
In general, boundary conditions are related to the coordinates describing the problem

28. Erwin Schrödinger 1887 – 1961 American physicist
Best known as one of the creators of quantum mechanics
His approach was shown to be equivalent to Heisenberg’s
Also worked with:
statistical mechanics
color vision
general relativity

29. Schrödinger Equation
The Schrödinger equation as it applies to a particle of mass m confined to moving along the x axis and interacting with its environment through a potential energy function U(x) is
This is called the time-independent Schrödinger equation

30. Schrödinger Equation, cont.
Both for a free particle and a particle in a box, the first term in the Schrödinger equation reduces to the kinetic energy of the particle multiplied by the wave function
Solutions to the Schrödinger equation in different regions must join smoothly at the boundaries

31. Schrödinger Equation, final
ψ(x) must be continuous
dψ/dx must also be continuous for finite values of the potential energy

32. Solutions of the Schrödinger Equation
Solutions of the Schrödinger equation may be very difficult
The Schrödinger equation has been extremely successful in explaining the behavior of atomic and nuclear systems
Classical physics failed to explain this behavior
When quantum mechanics is applied to macroscopic objects, the results agree with classical physics

33. Potential Wells
A potential well is a graphical representation of energy
The well is the upward-facing region of the curve in a potential energy diagram
The particle in a box is sometimes said to be in a square well
Due to the shape of the potential energy diagram

34. Schrödinger Equation Applied to a Particle in a Box
In the region 0 < x < L, where U = 0, the Schrödinger equation can be expressed in the form
The most general solution to the equation is ψ(x) = A sin kx + B cos kx
A and B are constants determined by the boundary and normalization conditions

35. Schrödinger Equation Applied to a Particle in a Box, cont.
Solving for the allowed energies gives
The allowed wave functions are given by
These match the original results for the particle in a box

36. Finite Potential Well A finite potential well is pictured
The energy is zero when the particle is 0 < x < L
In region II
The energy has a finite value outside this region
Regions I and III

37. Classical vs. Quantum Interpretation
According to Classical Mechanics
If the total energy E of the system is less than U, the particle is permanently bound in the potential well
If the particle were outside the well, its kinetic energy would be negative
An impossibility
According to Quantum Mechanics
A finite probability exists that the particle can be found outside the well even if E < U
The uncertainty principle allows the particle to be outside the well as long as the apparent violation of conservation of energy does not exist in any measurable way

38. Finite Potential Well – Region II
U = 0
The allowed wave functions are sinusoidal
The boundary conditions no longer require that ψ be zero at the ends of the well
The general solution will be
ψII(x) = F sin kx + G cos kx
where F and G are constants

39. Finite Potential Well – Regions I and III
The Schrödinger equation for these regions may be written as
The general solution of this equation is
A and B are constants

40. Finite Potential Well – Regions I and III, cont.
In region I, B = 0
This is necessary to avoid an infinite value for ψ for large negative values of x
In region III, A = 0
This is necessary to avoid an infinite value for ψ for large positive values of x
The solutions of the wave equation become

41. Finite Potential Well – Graphical Results for ψ
The wave functions for various states are shown
Outside the potential well, classical physics forbids the presence of the particle
Quantum mechanics shows the wave function decays exponentially to approach zero

42. Finite Potential Well – Graphical Results for ψ2
The probability densities for the lowest three states are shown
The functions are smooth at the boundaries

43. Active Figure 41.7
Use the active figure to adjust the length of the box
See the effect on the quantized states
PLAY
ACTIVE FIGURE

44. Finite Potential Well – Determining the Constants
The constants in the equations can be determined by the boundary conditions and the normalization condition
The boundary conditions are

45. Application – Nanotechnology
Nanotechnology refers to the design and application of devices having dimensions ranging from 1 to 100 nm
Nanotechnology uses the idea of trapping particles in potential wells
One area of nanotechnology of interest to researchers is the quantum dot
A quantum dot is a small region that is grown in a silicon crystal that acts as a potential well

46. Tunneling
The potential energy has a constant value U in the region of width L and zero in all other regions
This a called a square barrier
U is the called the barrier height

47. Tunneling, cont. Classically, the particle is reflected by the barrier
Regions II and III would be forbidden
According to quantum mechanics, all regions are accessible to the particle
The probability of the particle being in a classically forbidden region is low, but not zero
According to the uncertainty principle, the particle can be inside the barrier as long as the time interval is short and consistent with the principle

48. Tunneling, final
The curve in the diagram represents a full solution to the Schrödinger equation
Movement of the particle to the far side of the barrier is called tunneling or barrier penetration
The probability of tunneling can be described with a transmission coefficient, T, and a reflection coefficient, R

49. Tunneling Coefficients
The transmission coefficient represents the probability that the particle penetrates to the other side of the barrier
The reflection coefficient represents the probability that the particle is reflected by the barrier
T + R = 1
The particle must be either transmitted or reflected
T  e-2CL and can be nonzero
Tunneling is observed and provides evidence of the principles of quantum mechanics

50. Applications of Tunneling
Alpha decay
In order for the alpha particle to escape from the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleus-alpha particle system
Nuclear fusion
Protons can tunnel through the barrier caused by their mutual electrostatic repulsion

51. More Applications of Tunneling – Scanning Tunneling Microscope
An electrically conducting probe with a very sharp edge is brought near the surface to be studied
The empty space between the tip and the surface represents the “barrier”
The tip and the surface are two walls of the “potential well”

52. Scanning Tunneling Microscope
The STM allows highly detailed images of surfaces with resolutions comparable to the size of a single atom
At right is the surface of graphite “viewed” with the STM

53. Scanning Tunneling Microscope, final
The STM is very sensitive to the distance from the tip to the surface
This is the thickness of the barrier
STM has one very serious limitation
Its operation is dependent on the electrical conductivity of the sample and the tip
Most materials are not electrically conductive at their surfaces
The atomic force microscope overcomes this limitation

54. More Applications of Tunneling – Resonant Tunneling Device
The gallium arsenide in the center is a quantum dot
It is located between two barriers formed by the thin extensions of aluminum arsenide

55. Resonance Tunneling Devices, cont
Figure b shows the potential barriers and the energy levels in the quantum dot
The electron with the energy shown encounters the first barrier, it has no energy levels available on the right side of the barrier
This greatly reduces the probability of tunneling

56. Resonance Tunneling Devices, final
Applying a voltage decreases the potential with position
The deformation of the potential barrier results in an energy level in the quantum dot
The resonance of energies gives the device its name

57. Active Figure 41.11 Use the active figure to vary the voltage PLAY

58. Resonant Tunneling Transistor
This adds a gate electrode at the top of the resonant tunneling device over the quantum dot
It is now a resonant tunneling transistor
There is no resonance
Applying a small voltage reestablishes resonance

59. Simple Harmonic Oscillator
Reconsider black body radiation as vibrating charges acting as simple harmonic oscillators
The potential energy is
U = ½ kx2 = ½ mω2x2
Its total energy is
E = K + U = ½ kA2 = ½ mω2A2

60. Simple Harmonic Oscillator, 2
The Schrödinger equation for this problem is
The solution of this equation gives the wave function of the ground state as

61. Simple Harmonic Oscillator, 3
The remaining solutions that describe the excited states all include the exponential function
The energy levels of the oscillator are quantized
The energy for an arbitrary quantum number n is En = (n + ½)w where n = 0, 1, 2,…

62. Energy Level Diagrams – Simple Harmonic Oscillator
The separation between adjacent levels are equal and equal to DE = w
The energy levels are equally spaced
The state n = 0 corresponds to the ground state
The energy is Eo = ½ hω
Agrees with Planck’s original equations