Modern Physics Quantum Theory Limits of Classical Physics

History: Blackbody Radiation All bodies continuously radiate electro-magnetic waves. Blackbody – absorbs and remits all the electro- magnetic radiation that hits it. At a given temperature, the intensities of the electromagnetic waves emitted vary by wavelength Blackbody Simulation Copywrited by Holt, Rinehart, & Winston Wilson,Buffa, College Physics, 4 th ed 2000 Prentice Hall

History: Blackbody Radiation Blackbody – Ideal radiator Measuring the intensity of the electromagnetic radiation. – –The peak wavelength increases linearly with the temperature. – –This means that the temperature of a blackbody can be determined by its color. Classical physics calculations were completely unable to produce this temperature dependence, known as the “ultraviolet catastrophe.” Copyright ©2007 Pearson Prentice Hall, Inc.

Blackbody Radiation Problem – Classical calculated curve did not match the measured curve. To obtain agreement, Planck made the assumption that the electromagnetic energy is emitted only in discrete bundles (quanta) n=0,1,2,3… n=0,1,2,3… h is a constant known as Planck’s Constant. h = 6.63 x Js Planck Copywrited by Holt, Rinehart, & Winston E = nhf

Photoelectric Effect In 1888 Hallwachs observed a negatively charged zinc plate connected to an electroscope lost its charge when illuminated with ultra violet radiation

Photoelectric Effect Found that as light source is brought closer to a photo cell a larger photo current (I p ) is produced. The intensity of light is So, photocurrent is directly proportional to the intensity of light. From this a large current could be due to either: –Large number of electrons being emitted per second –Higher speed electrons being emitted.

Wilson,Buffa, College Physics, 4 th ed 2000 Prentice Hall Photoelectric Effect Milikan further investigated using the following setup: Variable voltage source used to create an opposing current. Voltage adjusted to make the current in the circuit zero. Known as the stopping voltage (V s ).

Photoelectric Effect Observed that: Stopping voltage stays the same regardless of the intensity of the light source. –Increase in current due to more electrons being emitted Stopping voltage depends on the frequency of the light source. Tsokos,Physics for the IB Diploma, Cambridge University Press 2005 Wilson,Buffa, College Physics, 4 th ed 2000 Prentice Hall

Photoelectric Effect Plotting the kinetic energy of the electrons versus frequency shows: A cutoff frequency or threshold frequency below which no electrons are emitted regardless of the intensity of light. Wilson,Buffa, College Physics, 4 th ed 2000 Prentice Hall

Photoelectric Effect These observations led to: The Laws of Photo-Electric Emission Number of electrons emitted per second is directly proportional to the intensity of the light. The maximum kinetic energy of emitted electrons increases with the frequency of the light. There is a minimum frequency below which no electrons are emitted.

Photoelectric Quandary Photoelectric Observations Intensity of light has no effect on the energy of the emitted electrons Electron energy depends on frequency of the light source, exists a minimum frequency below which no electrons are emitted. Classical Physics More intense beam of light contains more energy, so more energetic electrons should be emitted. No explanation for why the frequency affects the electron energy nor why there is a critical frequency.

Photoelectric Quandary Wave Particle Duality Copywrited by Holt, Rinehart, & Winston

Particle Nature of Light Einstein in 1905 proposed that light consists of quanta – packets of energy and momentum (photon) Energy of a photon is where E = energy (Joules), f = frequency (Hertz) h is Planck’s Constant. E = hf h = 6.63 x Js Copyright ©2007 Pearson Prentice Hall, Inc.

Particle Nature of Light Minimum amount of energy is needed to free an electron – work function of the metal ( ϕ ) If an electron has more energy than the work function, the additional energy will become the kinetic energy of the free electron. Tsokos,Physics for the IB Diploma, Cambridge University Press 2005

Particle Nature of Light From this result If E k vs f is graphed, a straight line with a slope of h is generated This matches the result of the photoelectric experiment. Wilson,Buffa, College Physics, 4 th ed 2000 Prentice Hall

Momentum of a Photon A photon is a relativistic particle – –It does not have mass, – –it does have momentum through Einstein’s relationship E=mc 2 – –the theory of relativity implies that it always travel at the speed of light (v=c). Recall, With  Since E=hf for a photon and with the wave equation v=fλ h = 6.63 x Js λ is the wavelength Compton Scattering Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Wave Nature of Matter Louis de Broglie in 1923 (as a grad student) – –Proposed that since light waves could exhibit particle-like behavior, particles of matter should exhibit wave-like behavior. Duality of matter. de Broglie Wavelength – –h is Planck’s constant – –p is the relativistic momentum of the particle. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Wave Nature of Matter Example: Determine the de Broglie wavelength for an electron (mass = 9.1 x kg) moving at a speed of 6.0 x 10 6 m/s and a baseball (mass = 0.15 kg) moving at a speed of 13 m/s. While all moving particle have a de Broglie wavelength, the effect of this wavelength are observable only for particles whose masses are very small.

Wave Nature of Matter de Broglie’s hypothesis was confirmed by an experiment by Clinton J Davisson and Lester H. Germer and independently by George P. Thomson Davisson-Germer experiment – –Directed a beam of electrons onto a crystal of nickel – –Electrons diffracted (wave behavior) and the wavelength shown by the diffraction pattern matched the one predicted by de Broglie’s hypothesis.

Wave Nature of Matter Wave Review – Diffraction A λ will diffract around an obstacle of size d only if the λ is comparable to or larger than d. The typical electron’s λ is ~ 7.2 x m. In a crystal the atoms are spaced ~ m. So a beam of electrons directed at a crystal lattice would form a diffraction pattern only if the crystal has inter-atomic spacing with similar dimensions to the de Broglie wavelength. Copyright ©2007 Pearson Prentice Hall, Inc.

Wave Nature of Matter If an electron passes through a double-slit arrangement (Young’s type), an interference pattern is observed beyond the two slits. Occurs even if one electron goes through the slits at each time. As if the electron ‘knows’ of the existence of both slits. As more electrons are sent through, bright fringes occur where there is a high probability of electrons striking the screen – true also for photons Particle waves are waves of probability. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Uncertainty Principle The extent of diffraction depends on the angle θ. To reach locations above or below the principal axis the electron must acquire some momentum in the y-axis. This is true despite the fact that the electron enters only moving in the x direction with no momentum in the y direction. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Uncertainty Principle The figure shows that the y component of the momentum may be as large as Δp y. Δp y represents the uncertainty in the y component of the momentum – –may have any value from 0 to Δp y. can be related to the width, W of the slit by applying wave diffraction theory. If small angle Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Uncertainty Principle p x is the x component of the electron’s momentum. From de Broglie’s equation So, A smaller slit width leads to larger uncertainty, Δp y Heisenberg suggested that Δp y is related to the uncertainty in the y position, Δy, of the electron as the electron passes through the slit. Since the electron can pass anywhere over the width of the slit, the uncertainty of y position is equal to the width. Δy=W 

Heisenberg Uncertainty Principle Limits the accuracy which with momentum and position can be known simultaneously. Fundamental limits imposed by nature. If the position is known Δy is zero so therefore Δp y must be infinitely large. Impossible to know both position and momentum of a particle at the same time. Similar relationship found between energy and time. ΔE uncertainty of the energy of the particle. Δt time interval at which particle remains in a given energy state. Uncertainty

Heisenberg Uncertainty Principle If the position of an object is know precisely so that the uncertainty in the position is only Δy = 1.5 x m. (a) Determine the minimum uncertainty in the momentum of the object. Find the corresponding minimum uncertainty in the speed of the object, if the object is an (a) electron (m = 9.1x kg) (b) a Ping-Pong ball (m = 2.2 x kg)

The ‘Electron in a Box’ Model An idealized situation of an electron in a box with infinitely high walls. Applying Schrodinger equation ψ(x,t) provides insight into the quantum theory. The electron treated as a wave has a wavelength associated with it defined by de Broglie as

The ‘Electron in the Box’ Model Since the electron cannot escape from the box, the electron wave can be assumed to be zero at the edges of the box. The electron cannot lose energy. Assume that the wave associated with the electron must be a standing wave. So the standing waves must have nodes at the walls or at x=0 and x=L. Implies, the wavelength must be related to the size of the box L by

The ‘Electron in a Box’ Model So using de Broglie’s relationship the momentum of the electron is The kinetic energy is Substituting, Our results show that the electron’s energy is quantized or discrete.

The ‘Electron in a Box’ Model Though oversimplified, the ‘electron in a box’ model indicates some important things about bound states for particles: The energies are quantized and can be characterized by a quantum number n The energy cannot be exactly zero The smaller the confinement, the larger the energy required. Source:

Quantum Tunneling Possible in microscopic situations Particle’s wavefunction is continuous It does not fall to zero immediately at the energy barrier, but it amplitude falls exponentially. If barrier is of a finite width then the wavefunction can continue on the other side. Low probability that the particle will escape even though it doesn’t have the energy to.

Quantum Theory Uncertainty Principle –If the momentum (velocity) is known, the position is unknown –If the position is known, the momentum (velocity) is unknown –Similar relationship between energy and time Quantized –Only discrete wavelengths can fit in the boundary conditions of a standing wave. Quantum PhysicsQuantum Theory