Problems remained from classical mechanics that the special theory of relativity didn’t explain

Problems remained from classical mechanics that the special theory of relativity didn’t explain.

• Problems remained from classical mechanics that the special theory of relativity didn’t explain.

• Attempts to apply the laws of classical physics to explain the behavior of matter on the atomic scale were consistently unsuccessful.

• Problems included:

• Blackbody radiation
• The electromagnetic radiation emitted by a heated object
• Photoelectric effect
• Emission of electrons by an illuminated metal

Between 1900 and 1930, another revolution took place in physics.

• Between 1900 and 1930, another revolution took place in physics.

• A new theory called quantum mechanics was successful in explaining the behavior of particles of microscopic size.

• The first explanation using quantum theory was introduced by Max Planck.

• Many other physicists were involved in other subsequent developments

An object at any temperature is known to emit thermal radiation.

• An object at any temperature is known to emit thermal radiation.

• Characteristics depend on the temperature and surface properties.
• The thermal radiation consists of a continuous distribution of wavelengths from all portions of the em spectrum.

• At room temperature, the wavelengths of the thermal radiation are mainly in the infrared region.

• As the surface temperature increases, the wavelength changes.

• It will glow red and eventually white.

The basic problem was in understanding the observed distribution in the radiation emitted by a black body.

• The basic problem was in understanding the observed distribution in the radiation emitted by a black body.

• Classical physics didn’t adequately describe the observed distribution.

• A black body is an ideal system that absorbs all radiation incident on it.

• The electromagnetic radiation emitted by a black body is called blackbody radiation.

A good approximation of a black body is a small hole leading to the inside of a hollow object.

• A good approximation of a black body is a small hole leading to the inside of a hollow object.

• The hole acts as a perfect absorber.

• The nature of the radiation leaving the cavity through the hole depends only on the temperature of the cavity.

The total power of the emitted radiation increases with temperature.

• The total power of the emitted radiation increases with temperature.

• Stefan’s law (from Chapter 20):
• P = A e T4
• The emissivity, e, of a black body is 1, exactly

• The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases.

• Wien’s displacement law
• maxT = 2.898 x 10-3 m . K

The intensity increases with increasing temperature.

• The intensity increases with increasing temperature.

• The amount of radiation emitted increases with increasing temperature.

• The area under the curve

• The peak wavelength decreases with increasing temperature.

An early classical attempt to explain blackbody radiation was the Rayleigh-Jeans law.

• An early classical attempt to explain blackbody radiation was the Rayleigh-Jeans law.

• At long wavelengths, the law matched experimental results fairly well.

At short wavelengths, there was a major disagreement between the Rayleigh-Jeans law and experiment.

• At short wavelengths, there was a major disagreement between the Rayleigh-Jeans law and experiment.

• This mismatch became known as the ultraviolet catastrophe.

• You would have infinite energy as the wavelength approaches zero.

1858 – 1847

• 1858 – 1847

• German physicist

• Introduced the concept of “quantum of action”

• In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy.

In 1900 Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation.

• In 1900 Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation.

• This equation is in complete agreement with experimental observations.

• He assumed the cavity radiation came from atomic oscillations in the cavity walls.

• Planck made two assumptions about the nature of the oscillators in the cavity walls.

The energy of an oscillator can have only certain discrete values En.

• The energy of an oscillator can have only certain discrete values En.

• En = n h ƒ
• n is a positive integer called the quantum number
• ƒ is the frequency of oscillation
• h is Planck’s constant
• This says the energy is quantized.
• Each discrete energy value corresponds to a different quantum state.
• Each quantum state is represented by the quantum number, n.

The oscillators emit or absorb energy when making a transition from one quantum state to another.

• The oscillators emit or absorb energy when making a transition from one quantum state to another.

• The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation.
• An oscillator emits or absorbs energy only when it changes quantum states.
• The energy carried by the quantum of radiation is E = h ƒ.

An energy-level diagram shows the quantized energy levels and allowed transitions.

• An energy-level diagram shows the quantized energy levels and allowed transitions.

• Energy is on the vertical axis.

• Horizontal lines represent the allowed energy levels.

• The double-headed arrows indicate allowed transitions.

The average energy of a wave is the average energy difference between levels of the oscillator, weighted according to the probability of the wave being emitted.

• The average energy of a wave is the average energy difference between levels of the oscillator, weighted according to the probability of the wave being emitted.

• This weighting is described by the Boltzmann distribution law and gives the probability of a state being occupied as being proportional to

• where E is the energy of the state.

Planck generated a theoretical expression for the wavelength distribution.

• Planck generated a theoretical expression for the wavelength distribution.

• h = 6.626 x 10-34 J.s
• h is a fundamental constant of nature.

• At long wavelengths, Planck’s equation reduces to the Rayleigh-Jeans expression.

• At short wavelengths, it predicts an exponential decrease in intensity with decreasing wavelength.

• This is in agreement with experimental results.

Einstein rederived Planck’s results by assuming the oscillations of the electromagnetic field were themselves quantized.

• Einstein rederived Planck’s results by assuming the oscillations of the electromagnetic field were themselves quantized.

• In other words, Einstein proposed that quantization is a fundamental property of light and other electromagnetic radiation.

• This led to the concept of photons.

The photoelectric effect occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces.

• The photoelectric effect occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces.

• The emitted electrons are called photoelectrons.
• They are no different than other electrons.
• The name is given because of their ejection from a metal by light in the photoelectric effect

When the tube is kept in the dark, the ammeter reads zero.

• When the tube is kept in the dark, the ammeter reads zero.

• When plate E is illuminated by light having an appropriate wavelength, a current is detected by the ammeter.

• The current arises from photoelectrons emitted from the negative plate and collected at the positive plate.

At large values of V, the current reaches a maximum value.

• At large values of V, the current reaches a maximum value.

• All the electrons emitted at E are collected at C.

• The maximum current increases as the intensity of the incident light increases.

• When V is negative, the current drops.

• When V is equal to or more negative than Vs, the current is zero.

Dependence of photoelectron kinetic energy on light intensity

• Dependence of photoelectron kinetic energy on light intensity

• Classical Prediction
• Electrons should absorb energy continually from the electromagnetic waves.
• As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy.
• Experimental Result
• The maximum kinetic energy is independent of light intensity.
• The maximum kinetic energy is proportional to the stopping potential (Vs).

Time interval between incidence of light and ejection of photoelectrons

• Time interval between incidence of light and ejection of photoelectrons

• Classical Prediction
• At low light intensities, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal.
• This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal.
• Experimental Result
• Electrons are emitted almost instantaneously, even at very low light intensities.

Dependence of ejection of electrons on light frequency

• Dependence of ejection of electrons on light frequency

• Classical Prediction
• Electrons should be ejected at any frequency as long as the light intensity is high enough.
• Experimental Result
• No electrons are emitted if the incident light falls below some cutoff frequency, ƒc.
• The cutoff frequency is characteristic of the material being illuminated.
• No electrons are ejected below the cutoff frequency regardless of intensity.

Dependence of photoelectron kinetic energy on light frequency

• Dependence of photoelectron kinetic energy on light frequency

• Classical Prediction
• There should be no relationship between the frequency of the light and the electric kinetic energy.
• The kinetic energy should be related to the intensity of the light.
• Experimental Result
• The maximum kinetic energy of the photoelectrons increases with increasing light frequency.

The experimental results contradict all four classical predictions.

• The experimental results contradict all four classical predictions.

• Einstein extended Planck’s concept of quantization to electromagnetic waves.

• All electromagnetic radiation of frequency ƒ from any source can be considered a stream of quanta, now called photons.

• Each photon has an energy E and moves at the speed of light in a vacuum.

• E = hƒ

• A photon of incident light gives all its energy to a single electron in the metal.

Electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy Kmax.

• Electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy Kmax.

• Kmax = hƒ – φ

• φ is called the work function of the metal.
• The work function represents the minimum energy with which an electron is bound in the metal.

Dependence of photoelectron kinetic energy on light intensity

• Dependence of photoelectron kinetic energy on light intensity

• Kmax is independent of light intensity.
• K depends on the light frequency and the work function.

• Time interval between incidence of light and ejection of the photoelectron

• Each photon can have enough energy to eject an electron immediately.

• Dependence of ejection of electrons on light frequency

• There is a failure to observe photoelectric effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron.
• Without enough energy, an electron cannot be ejected, regardless of the fact that many photons per unit time are incident on the metal in a very intense light beam.

Dependence of photoelectron kinetic energy on light frequency

• Dependence of photoelectron kinetic energy on light frequency

• Since Kmax = hƒ – φ
• A photon of higher frequency carries more energy.
• A photoelectron is ejected with higher kinetic energy.
• Once the energy of the work function is exceeded
• There is a linear relationship between the maximum electron kinetic energy and the frequency.

The lines show the linear relationship between K and ƒ.

• The lines show the linear relationship between K and ƒ.

• The slope of each line is h.

• The x-intercept is the cutoff frequency.

• This is the frequency below which no photoelectrons are emitted.

The cutoff frequency is related to the work function through ƒc = φ / h.

• The cutoff frequency is related to the work function through ƒc = φ / h.

• The cutoff frequency corresponds to a cutoff wavelength.

• Wavelengths greater than c incident on a material having a work function φ do not result in the emission of photoelectrons.

1892 – 1962

• 1892 – 1962

• American physicist

• Director of the lab at the University of Chicago

• Discovered the Compton Effect

• Shared the Nobel Prize in 1927

Compton and Debye extended Einstein’s idea of photon momentum.

• Compton and Debye extended Einstein’s idea of photon momentum.

• The two groups of experimenters accumulated evidence of the inadequacy of the classical wave theory.

• The classical wave theory of light failed to explain the scattering of x-rays from electrons.

According to the classical theory, em waves incident on electrons should:

• According to the classical theory, em waves incident on electrons should:

• Have radiation pressure that should cause the electrons to accelerate
• Set the electrons oscillating
• There should be a range of frequencies for the scattered electrons.

Compton’s experiments showed that, at any given angle, only one frequency of radiation is observed.

• Compton’s experiments showed that, at any given angle, only one frequency of radiation is observed.

The results could be explained by treating the photons as point-like particles having energy hƒ and momentum h ƒ / c.

• The results could be explained by treating the photons as point-like particles having energy hƒ and momentum h ƒ / c.

• Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved.

• This scattering phenomena is known as the Compton effect.

The graphs show the scattered x-rays for various angles.

• The graphs show the scattered x-rays for various angles.

• The shifted peak, λ’, is caused by the scattering of free electrons.

• This is called the Compton shift equation.

The factor h/mec in the equation is called the Compton wavelength of the electron and is

• The factor h/mec in the equation is called the Compton wavelength of the electron and is

• The unshifted wavelength, λo, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms.

Some experiments are best explained by the photon model.

• Some experiments are best explained by the photon model.

• Some are best explained by the wave model.

• We must accept both models and admit that the true nature of light is not describable in terms of any single classical model.

• The particle model and the wave model of light complement each other.

• A complete understanding of the observed behavior of light can be attained only if the two models are combined in a complementary matter.

1892 – 1987

• 1892 – 1987

• French physicist

• Originally studied history

• Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons

Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties.

• Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties.

• The de Broglie wavelength of a particle is

In an analogy with photons, de Broglie postulated that a particle would also have a frequency associated with it

• In an analogy with photons, de Broglie postulated that a particle would also have a frequency associated with it

• These equations present the dual nature of matter:

• Particle nature, p and E
• Wave nature, λ and ƒ

The principle of complementarity states that the wave and particle models of either matter or radiation complement each other.

• The principle of complementarity states that the wave and particle models of either matter or radiation complement each other.

• Neither model can be used exclusively to describe matter or radiation adequately.

If particles have a wave nature, then under the correct conditions, they should exhibit diffraction effects.

• If particles have a wave nature, then under the correct conditions, they should exhibit diffraction effects.

• Davisson and Germer measured the wavelength of electrons.

• This provided experimental confirmation of the matter waves proposed by de Broglie.

Mechanical waves have materials that are “waving” and can be described in terms of physical variables.

• Mechanical waves have materials that are “waving” and can be described in terms of physical variables.

• A string may be vibrating.
• Sound waves are produced by molecules of a material vibrating.
• Electromagnetic waves are associated with electric and magnetic fields.

• Waves associated with particles cannot be associated with a physical variable.

The electron microscope relies on the wave characteristics of electrons.

• The electron microscope relies on the wave characteristics of electrons.

• Shown is a transmission electron microscope

• Used for viewing flat, thin samples

• The electron microscope has a high resolving power because it has a very short wavelength.

• Typically, the wavelengths of the electrons are about 100 times shorter than that of visible light.

The quantum particle is a new model that is a result of the recognition of the dual nature of both light and material particles.

• The quantum particle is a new model that is a result of the recognition of the dual nature of both light and material particles.

• Entities have both particle and wave characteristics.

• We must choose one appropriate behavior in order to understand a particular phenomenon.

An ideal particle has zero size.

• An ideal particle has zero size.

• Therefore, it is localized in space.

• An ideal wave has a single frequency and is infinitely long.

• Therefore, it is unlocalized in space.

• A localized entity can be built from infinitely long waves.

Multiple waves are superimposed so that one of its crests is at x = 0.

• Multiple waves are superimposed so that one of its crests is at x = 0.

• The result is that all the waves add constructively at x = 0.

• There is destructive interference at every point except x = 0.

• The small region of constructive interference is called a wave packet.

• The wave packet can be identified as a particle.

The dashed line represents the envelope function.

• The dashed line represents the envelope function.

• This envelope can travel through space with a different speed than the individual waves.

The phase speed of a wave in a wave packet is given by

• The phase speed of a wave in a wave packet is given by

• This is the rate of advance of a crest on a single wave.

• The group speed is given by

• This is the speed of the wave packet itself.

The group speed can also be expressed in terms of energy and momentum.

• The group speed can also be expressed in terms of energy and momentum.

• This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent.

Parallel beams of mono-energetic electrons that are incident on a double slit.

• Parallel beams of mono-energetic electrons that are incident on a double slit.

• The slit widths are small compared to the electron wavelength.

• An electron detector is positioned far from the slits at a distance much greater than the slit separation.

If the detector collects electrons for a long enough time, a typical wave interference pattern is produced.

• If the detector collects electrons for a long enough time, a typical wave interference pattern is produced.

• This is distinct evidence that electrons are interfering, a wave-like behavior.

• The interference pattern becomes clearer as the number of electrons reaching the screen increases.

A maximum occurs when

• A maximum occurs when

• This is the same equation that was used for light.

• This shows the dual nature of the electron.

• The electrons are detected as particles at a localized spot at some instant of time.
• The probability of arrival at that spot is determined by finding the intensity of two interfering waves.

An electron interacts with both slits simultaneously.

• An electron interacts with both slits simultaneously.

• If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern.

• It is impossible to determine which slit the electron goes through.

• In effect, the electron goes through both slits.

• The wave components of the electron are present at both slits at the same time.

1901 – 1976

• 1901 – 1976

• German physicist

• Developed matrix mechanics

• Many contributions include:

• Uncertainty principle
• Received Nobel Prize in 1932
• Prediction of two forms of molecular hydrogen
• Theoretical models of the nucleus

In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty.

• In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty.

• Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy.

• The Heisenberg uncertainty principle states: if a measurement of the position of a particle is made with uncertainty x and a simultaneous measurement of its x component of momentum is made with uncertainty px, the product of the two uncertainties can never be smaller than /2.

It is physically impossible to measure simultaneously the exact position and exact momentum of a particle.

• It is physically impossible to measure simultaneously the exact position and exact momentum of a particle.

• The inescapable uncertainties do not arise from imperfections in practical measuring instruments.

• The uncertainties arise from the quantum structure of matter.

Another form of the uncertainty principle can be expressed in terms of energy and time.

• Another form of the uncertainty principle can be expressed in terms of energy and time.

• This suggests that energy conservation can appear to be violated by an amount E as long as it is only for a short time interval t.

The Uncertainty Principle cannot be interpreted as meaning that a measurement interferes with the system.

• The Uncertainty Principle cannot be interpreted as meaning that a measurement interferes with the system.

• The Uncertainty Principle is independent of the measurement process.

• It is based on the wave nature of matter.