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Chapter 41

Chapter 41. Quantum Mechanics. Quantum Mechanics. The theory of quantum mechanics was developed in the 1920s By Erwin Schr ödinger, Werner Heisenberg and others Enables use to understand various phenomena involving Atoms, molecules, nuclei and solids. Probability – A Particle Interpretation.

theodore-gilliam
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Chapter 41

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  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|2dx • 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. 41.3 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

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