1 / 36

# Chapter41 - PowerPoint PPT Presentation

Chapter 41 Quantum Mechanics de Broglie’s Idea Particles had wave characteristics just like waves had particle characteristics (e.g. E&M wave  photon) The de Broglie wavelength of a particle of mass, m, is The frequency of matter waves is Uncertainty Principle and wave character

Related searches for Chapter41

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Chapter41' - issac

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Chapter 41

Quantum Mechanics

• Particles had wave characteristics just like waves had particle characteristics (e.g. E&M wave  photon)

• The de Broglie wavelength of a particle of mass, m, is

• The frequency of matter waves is

• Mathematically,

• For a wave with λ one has Δx~λ and by deBroglie Δp~h/λ

• Thus ΔxΔp~h

• All things have a dual particle/wave nature. Which property is important/noticeable depends on the experiment

• The light two slit experiment reveals wave nature of light

• The Compton experiment, photoelectric effect reveals the particle nature of light

• De Broglie’s hypothesis suggested a wave character for particles.

• This wave nature revealed itself in the Davisson-Germer experiment.

• Also in the particle two-slit experiment

• Which slit does a particle go through?

• The answer is both!  Probability wave

• What are these matter waves of?

• They are probability waves described by a complex valued function of x and t i.e. Ψ(x,t)

• The probability to find the particle in some dx is given

• Normalization

• Finite probability

• Expectation value (average)

• Classically the particle has equal probability to be anywhere in the box

• But if xi and pi are known then one knows exactly where the ball is at any later time

• Quantum mechanically L=nλ/2 or λ=2L/n (n=1,2,3…)

• The wave function (guess) ψ(x)=Asin(kx) with k=2π/λ=nπ /L

• Giving ψ(x)=Asin(n πx/L). A=√(2/L) by normalization

• The energy spectrum is found as follows

• De Broglie  p=h/λ=hn/2L (n=1, 2, 3, …)

• Energy En = p2 /2m = h2 n2 / (8 m L2 )

• n=0 not allowed. Lowest energy is n=1 the zero point energy. This is different from classical mechanics where E=0 is possible. This is connected with the Heisenberg uncertainty principle.

The Schrödinger Equation for ψ I

• In 1926 Schrödinger proposed a wave equation that describes the manner in which de Broglie matter waves change in space and time

• Schrödinger’s wave equation is a key element in quantum mechanics

• Schrödinger’s wave equation is generally solved for the wave function, Ψ

The Schrödinger Equation for ψ II

• The 1D Schrödinger Equation for a particle with energy E, moving in a potential U(x), having mass, m, is

• The full (complex) wave function is

• In terms of Schrödinger’s Equation the 1D box problem becomes

• Where

• Solutions

• Boundary Conditions give

The Schrödinger Cat Problem

• It’s OK for electrons and other small things but what about marcoscopic things like a cat?

• A gas at low pressure has a voltage applied to it

• A gas emits light characteristic of the gas

• When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed

• Each line has a different wavelength and color

• This series of lines is called an emission spectrum

• The wavelengths of hydrogen’s spectral lines can be found from

• RH is the Rydberg constant

• RH = 1.0973732 x 107 m-1

• n is an integer, n = 1, 2, 3, …

• The spectral lines correspond to different values of n

• The Balmer Series has lines whose wavelengths are given by the preceding equation

• Examples of spectral lines

• n = 3, λ = 656.3 nm

• n = 4, λ = 486.1 nm

• In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory

• His model includes both classical and non-classical ideas

• His model included an attempt to explain why the atom was stable

• The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction

• The Coulomb force produces the centripetal acceleration

• Only certain electron orbits are stable

• These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation

• Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion

• Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state

• The “jump” cannot be treated classically

• The electron’s “jump,” continued

• The frequency emitted in the “jump” is related to the change in the atom’s energy

• It is generally not the same as the frequency of the electron’s orbital motion

• The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum

• Electron’s orbital angular momentum

• me v r = n ħ where n = 1, 2, 3, …

• The total energy of the atom

• The energy can also be expressed as

• The radii of the Bohr orbits are quantized

• This shows that the electron can only exist in certain allowed orbits determined by the integer n

• When n = 1, the orbit has the smallest radius, called the Bohr radius, ao

• ao = 0.0529 nm

• A general expression for the radius of any orbit in a hydrogen atom is

• rn = n2 ao

• The energy of any orbit is

• En = - 13.6 eV/ n2

• The lowest energy state is called the ground state

• This corresponds to n = 1

• Energy is –13.6 eV

• The next energy level has an energy of –3.40 eV

• The energies can be compiled in an energy level diagram

• The ionization energy is the energy needed to completely remove the electron from the atom

• The ionization energy for hydrogen is 13.6 eV

• The value of RH from Bohr’s analysis is in excellent agreement with the experimental value

• A more generalized equation can be used to find the wavelengths of any spectral lines

• For the Balmer series, nf = 2

• For the Lyman series, nf = 1

• Whenever an transition occurs between a state, ni to another state, nf (where ni > nf), a photon is emitted

• The photon has a frequency f = (Ei – Ef)/h and wavelength λ

• Explained several features of the hydrogen spectrum

• Accounts for Balmer and other series

• Predicts a value for RH that agrees with the experimental value

• Gives an expression for the radius of the atom

• Predicts energy levels of hydrogen

• Gives a model of what the atom looks like and how it behaves

• Can be extended to “hydrogen-like” atoms

• Those with one electron

• Ze2 needs to be substituted for e2 in equations

• Z is the atomic number of the element

A non-relativistic electron and a non-relativistic proton are moving and have the same de Broglie wavelength. Which of the following are also the same for the two particles: (a) speed, (b) kinetic energy, (c) momentum, (d) frequency?

QUICK QUIZ 27.4

(c). Two particles with the same de Broglie wavelength will have the same momentum p = mv. If the electron and proton have the same momentum, they cannot have the same speed because of the difference in their masses. For the same reason, remembering that KE = p2/2m, they cannot have the same kinetic energy. Because the kinetic energy is the only type of energy an isolated particle can have, and we have argued that the particles have different energies, Equation 27.15 tells us that the particles do not have the same frequency.

We have seen two wavelengths assigned to the electron, the Compton wavelength and the de Broglie wavelength. Which is an actual physical wavelength associated with the electron: (a) the Compton wavelength, (b) the de Broglie wavelength, (c) both wavelengths, (d) neither wavelength?

QUICK QUIZ 27.5

(b). The Compton wavelength, λ Compton wavelength and the de Broglie wavelength. Which is an actual C = h/mec, is a combination of constants and has no relation to the motion of the electron. The de Broglie wavelength, λ= h/mev, is associated with the motion of the electron through its momentum.