Quantum Theory Glenn V. Lo Department of Physical Sciences Nicholls State University

Modern Physics Physics principles established before the end of the 19 th century are now called classical physics; “anomalous” experiments led to the development of modern physics. Failure of classical physics to explain phenomena dealing with the world of the very fast (approaching speed of light) led to the development of the theory of Relativity. Failure to explain phenomena dealing with the world of the very small led to the development of Quantum Mechanics.

Light Experiments dealing with light were critical to the development of modern atomic theory, which is based on Quantum Mechanics. Light is also known as “electromagnetic radiation”

Electromagnetic wave Behavior of light are well-explained by thinking of light as an electromagnetic wave. Wave = periodic disturbance propagating through space. “Periodic” means repeating. “Electromagnetic” means it affects charged particles. A wave transmits energy from one point in space to another.

Describing EM wave What’s waving? Electric and magnetic fields. Electric field (E) = force that a charge of 1 C would “feel” at a given location Electric field is strongest at crests and troughs. Force felt at crests and troughs are in opposite directions. No force felt at nodes. Magnitude of E at crests and troughs, E o, is called the “amplitude” of the wave. Magnetic field wave is perpendicular to electric field wave: also affects charged particles. EEoEo

Describing EM wave Speed (c) = 2.998x10 8 m/s Distance between consecutive crests (or troughs) is called the wavelength ( ). Frequency (represented by Greek letter nu, ) = number of cycles passing through a point per second; unit: s -1 or Hz Distance traveled per cycle TIMES cyles per second equals distance traveled per second: = c E

Example What is the frequency of green light with a wavelength of nm? Answer: 6.000x10 14 Hz

Interpreting Wavelength and Frequency Wavelength and Frequency = “color” Visible light: nm; 1 nm = m “Electromagnetic spectrum” = range of possible wavelengths. Most electromagnetic radiation is invisible to us. Longer wavelength x-rays, ultraviolet infrared, microwave, radiowave Lower frequency

Example Which color of light corresponds to longer wavelength? A. red, B. blue Which color of light corresponds to higher frequency? A. red, B. blue Which type of electromagnetic radiation corresponds to longer wavelength? A. microwave, B. x-ray Longer wavelength x-rays, ultraviolet infrared, microwave, radiowave Lower frequency

Interpreting Amplitude Amplitude = “E o ”, is related to the intensity (S) of the radiation; energy delivered per unit time per unit area or power delivered per unit area Unit for Intensity: J s -1 m -2, or W m -2  o = “epsilon naught” = permittivity of vacuum = 8.854x C 2 N -1 m -2 EEoEo

Example A 5.0 mW laser beam has a cross sectional area of 2.0 mm 2. What is the intensity of the beam? Calculate the magnitude of electric field wave.  o = 8.854x C 2 N -1 m -2, c = 2.998x10 8 m/s Answer: S = 2.5x10 3 W m -2, E o =1.37x10 3 N C -1

The Particulate Model The notion of light as an electromagnetic wave is referred to as a “wave model” Problem: Towards the end of 19 th century, several experimental phenomena involving interaction of light with atoms and molecules could not be explained by the wave model. Blackbody radiation Photoelectric effect Atomic and molecular spectra Solution: imagine light as a stream of particles, called “photons”

Photons Energy of photon = proportional to frequency of the light. Proportionality constant (h) is called “Planck’s constant” h = 6.626x J s Higher frequency  More energetic photons Example: which has more energetic photons, radiowave or x-ray?

Example Calculate the energy of each photon of red light with a wavelength of 632 nm. Answer: 3.14x J

Example The frequency of radiation used in a microwave oven is 2.45 GHz. If a W oven is used for 60.0 s, how many photons are generated? Answer: 2.22x10 28 J

eV Typical photon energies are very small; eV (electron volt) is a commonly used unit for convenience. 1 eV = 1.602x J A photon has an energy of 3.14x J. Express this value in eV. Answer: 1.96 eV

Test Yourself Illustrated below is a snapshot of the electric field component of an electromagnetic wave as it travels through space. At which of the locations indicated would a charged particle experience the strongest force?

Test Yourself What is the frequency of the electromagnetic wave with a wavelength of 250 nm? A. 1.2x10 6 Hz, B. 1.2x10 15 Hz, C. 6.0x10 14 Hz

Test Yourself Which of the following regions of the electromagnetic spectrum has the most energetic photons? A. x-rays, B. visible, C. microwave, D. radiowave

Test Yourself Which of the following characteristics of electromagnetic radiation is not associated with what we perceive as color? A. wavelength, B. frequency, C.wave speed, D. photon energy

Test Yourself What is the energy of a photon of ultraviolet radiation with a wavelength of 250 nm? A. 5.0 eV, B. 2.5 eV, C. 4.0x10 -2 eV

Test Yourself How many photons are produced by a 5.0 mW green laser beam ( =500.0 nm) in 20.0 s?

Photoelectric Effect: the facts Light can cause electrons to be ejected from metals. Depending on material, minimum (threshold) frequency ( o ) is needed for electrons to be ejected. “Anomalous” observation: the maximum kinetic energy of photoelectrons does not depend on intensity, but depends on the frequency Number of photoelectrons ejected depends on intensity.

Einstein’s explanation Imagine light as a stream of photons Each photon carries energy equal to h There is a minimum energy needed to dislodge an electron from the surface (  ). If h < , photon cannot dislodge electron If h > , KE of photoelectron = h -  Higher intensity means more photons. See: (start at 5:56)

Example A nm radiation has photon energy of eV. Al has a work function, , of 4.08 eV. What happens if we shine light of nm on an aluminum metal?

Atomic and Molecular Spectra Light produced by atoms can be separated into different wavelengths (just like a rainbow). What we observe is called an emission spectrum. Absorption spectrum is what we observe when we pass light through a sample, then separate the light that emerges into different wavelengths.

Atomic and Molecular Spectra A continuous spectrum is obtained from a light source with a high concentration of atoms (like a solid such as the filament of an incandenscent light bulb).

Emission Spectra When concentration of atoms in a light source is low (like in a gas sample), line spectrum is observed. Each atom/molecule has a unique spectrum. Example: emission spectrum of hydrogen atoms See (start at 2:22)

Absorption Spectra If a continuous spectrum is passed through a sample where the concentration of atoms is low (like in a gas sample), line spectra are also observed. Each atom or molecule has a unique spectrum. Examples: Hydrogen Sodium

Bohr’s “quantum” explanation of line spectra Energy of atoms and molecules are “quantized” or restricted (cannot have just any value) Energy of photon absorbed or released must match difference between allowed energy levels: E photon = E upper – E lower Since values for E upper and E lower are restricted, values for E photon also restricted. Since E photon = h = hc/, values of and are also restricted

Example Bohr’s model predicts that the two lowest allowed energy levels of hydrogen are eV and eV, respectively. What is the energy of the photon that will be absorbed by a hydrogen atom for it to be “excited” from eV to eV? What wavelength of light does this correspond to?  Answer: 10.2 eV, 122 nm

Diffraction Diffraction = bending of waves as they pass a barrier. Pattern observed when light is passed through a slit. Pattern spreads out more with narrower slit.

Explaining Diffraction As waves are bent, waves emerging at different points could end up being “out of sync” – leading to “destructive interference” (dark spots) Bright spots are due to “constructive interference” of waves

Wave-particle Duality Refers to the fact that light, which we considered as a wave, sometimes behaves as though it is a stream of particles. De Broglie (1920s): suggested that wave-particle duality also applies to particles like electrons. = h/p = h/(mv), p=momentum De Broglie’s hypothesis was confirmed by experiments of Davisson and Germer: electron diffraction; basis for electron microscope

Example Calculate the de Broglie wavelength of a 1.00 kg object moving at 1.00 m/s. Answer: 6.63x m; wavelike behavior is not detectable because the wavelength is much much smaller than the diameter of a nucleus, which is between – m

Example Calculate the de Broglie wavelength of an electron (mass = 9.11x kg) moving at a speed of 1.00x10 6 m/s. Answer: 7.27x m; comparable to typical distance between atoms in a solid (1-2 Angstroms; 1 Angstrom = m). Electrons will exhibit wavelike behavior (such as diffraction) when they encounter a solid.

Heisenberg’s Uncertainty Principle Consequence of wave-particle duality It is impossible to precisely determine, simultaneously, both position and momentum (or velocity) of an electron:  x  p > h/2  where  x = uncertainty in location  p = uncertainty in momentum p = mv h = Planck’s constant

Heisenberg’s Uncertainty Principle Pinpointing the location of an electron by bouncing light off it alters its motion since light behaves like a particle (photon). You can try to precisely locate electrons in a beam by passing the beam through a slit; beam diffracts like light, spreads out more the narrower you make the slit. Narrower slit  more precise location  less precise velocity

Quantum Theory Classical physics predicts a nuclear atom cannot be stable. Classical physics predicts that an electron going around the nucleus is accelerating; since an electron is electrically charged, as it accelerates, it:  will continually emit light  will continually lose energy  will eventually spiral and crash into the nucleus Bohr’s solution: assume there are stable orbits; energy is restricted (quantized) Problems: only works for one electron, cannot explain fine structure; violates Heisenberg’s uncertainty principle

Quantum Mechanics Heisenberg’s uncertainty principle suggests that classical prediction is wrong. Nature does not allow us to predict exactly how the electron will move. Nuclear model requires a completely new theory (Quantum Mechanics). QM is a probabilistic theory; classical mechanics is deterministic

Test Yourself If a metallic surface with a work function of 2.3 eV is exposed to light, electrons will not be ejected if the photon energy is A. 2.1 eV, B. 2.5 eV

Test Yourself If electrons are not ejected from a metallic surface when exposed to blue light, we expect that exposure to red light would A. cause electrons to be ejected, B. not cause electrons to be ejected

Test Yourself If a metallic surface with a work function of 2.5 eV is exposed to light with photon energy of 2.8 eV, the maximum kinetic energy of the electrons is... A. 0.3 eV, B. 5.3 eV

Test Yourself An atom loses 2.0 eV of energy by emitting light. The energy of the photon released is... A. 1.0 eV, B. 2.0 eV, C. 4.0 eV

Test Yourself Will photoelectrons be ejected if we shine nm on a metal that has a work function of eV?

Test Yourself The three lowest allowed energy levels of a particle are 0.5 eV, 1.5 eV, and 2.5 eV. If the particle happens to have an energy of 0.5 eV, it will not absorb a photon with energy of... A. 0.5 eV, B. 1.0 eV, C. 2.0 eV

Test Yourself The lowest allowed energy levels of a particle are: a, 4a, 9a, and 16a. Which of these cannot possibly be the energy of a photon emitted by a particle with energy of 9a? A. 5a, B. 7a, C. 8a, D. 9a

Test Yourself Theory predicts that the allowed energies of hydrogen are given by the formula: E = (-13.6 eV) / n 2, where n=1, 2, 3,... Which of these photon energies cannot be absorbed by a hydrogen atom that has an energy of eV? A. 5.1 eV B eV C eV