Special Relativity and Quantum Physics Special Relativity and ...

Special Relativity and Quantum Physics

Special Relativity and Quantum Physics • Albert Einstein’s theory of special relativity describes this phenomenon correctly. It is based on two postulates:

1 – The Principle of Relativity • Although Newtonian mechanics works very well at low speeds, it fails when applied to particles whose speeds approach that of light.

1. The laws of physics are the same in all inertial reference frames.

• Experimentally, the predictions of Newtonian mechanics at high speeds can be tested by accelerating an electron through a large electric potential difference. For example, an electron can be accelerated to v = 0.99c by using a potential difference of several million volts.

2. The speed of light (in vacuum) has always the same value of c, and is independent of the motion of the observer or of the light source.

• According to Newtonian mechanics, the speed should be doubled to v = 1.98c, if the potential difference is increased by a factor 4.

• To describe a physical event, it is necessary to specify a frame of reference, i.e. a coordinate system and a clock. Reference frames in which Newton’s first law (law of inertia) is valid are called inertial frames.

• However, experiments show that the speed of the electron remains lower than the speed of light, no matter how large the potential difference is.

• Since the laws of physics are the same in all inertial reference frames, there is no preferred frame and there is no experiment that can distinguish between different inertial frames.

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2 – The Speed of Light

(2) the laws of electricity and magnetism are not the same in all inertial frames.

There is a serious contradiction between the Newtonian addition law for velocities and the fact that the speed of light is the same for all observers.

• If (1) is correct, notions of absolute time and length are incorrect.

= c is sent out by an observer • Suppose a light pulse with speed v relative to the ground. According in a truck moving with speed v to Newtonian addition of velocities, the light pulse has a speed v + > c relative to a stationary observer (?). v

• If (2) is correct, a preferred reference frame must exist.

• In the 19th century it was thought that electromagnetic waves require a medium to propagate, the so-called ether, which was to be present everywhere, even in empty space.

• If the ether actually existed, a preferred (“absolute”) frame would exist. If (!) the Sun is assumed to be at rest in the ether, and v is the velocity of the Earth with respect to the “ether”, the speed of light would then be c + v “downwind”, and c − v “upwind”. • The Michelson-Morley experiment was designed to detect these small changes in the speed of light.

• Therefore, either (1) the law for addition of velocities is wrong, or Dr.D.Wackeroth

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• Its result was negative, thus contradicting the ether hypothesis. PHY102A

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Special Relativity and Quantum Physics Using the Pythagorean theorem gives: 2 2 v∆t c∆t = + d2 2 2

3 – Consequences of Special Relativity • The Relativity of Time Consider a vehicle moving to the right with a speed v. A mirror is fixed to the ceiling of the vehicle and an observer in the vehicle holds a flash gun a distance d below the mirror. The time it takes a light flash to travel from the moving observer to the mirror and back again measured by the moving observer is

Solving for ∆t and substituting d = c∆t0 /2 one finds: ∆t = p

Now consider the same set of events as viewed by an observer at rest. For him, mirror and flash gun move with v to the right. When the light strikes the mirror, it has moved a distance v∆t/2 horizontally, and d vertically. Here ∆t is the time difference as measured by the observer at rest. Spring 2005

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4 – Relativistic Momentum • In order to properly describe the motion of relativistic particles, Newton’s 2nd law and the definitions of momentum and energy need to be modified.

Consider a spaceship traveling with speed v between two stars. The proper distance between the stars is L0 . For an observer at rest with respect to the stars, the time it takes for the trip is ∆t = L0 /v. The space traveler measures ∆t0 = ∆t/γ and the distance between the stars is ∆t L = v∆t0 = v γ Because L0 = v∆t, p L0 = L0 1 − v 2 /c2 γ

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• Length Contraction The measured distance between two points depends on the frame of reference. The proper length of an object is the length measured by an observer at rest with respect to the object.

L=

(1)

p where γ = 1/ 1 − v 2 /c2 . In words: The time interval measured by the observer in the stationary frame is longer than that measured by the observer in the moving frame, i.e. a moving clock runs more slowly than an identical stationary clock (time dilation). ∆t0 is called the proper time. The proper time is always the time measured by an observer moving along with the clock. Example for time dilation: Lifetime of cosmic muons

2d ∆t0 = c

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∆t0 = γ∆t0 1 − v 2 /c2

• Requiring momentum conservation and that the relativistic momentum approaches the “classical” value, m0 v, for v/c → 0, one can show that the relativistic momentum is given by p= p

m0 v = γm0 v 1 − v 2 /c2

(3)

where m0 is the rest mass of the particle. The rest mass is the mass measured by an observer at rest with respect to the particle.

(2)

In words: length contraction takes place along the direction of motion. Dr.D.Wackeroth

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Special Relativity and Quantum Physics 6 – Relativistic Addition of Velocities • Imagine a motorcycle rider moving with v = 0.8c past a stationary observer. If the rider tosses a ball with a speed of v = 0.7c in the forward direction relative to himself, what is the speed of the ball recorded by the stationary observer (v )?

5 – Mass and the Ultimate Speed

• Einstein also showed that the mass of an object depends on the frame of reference. It increases with speed according to

• Common sense tells us the answer is v =v +v = 0.8c + 0.7c = 1.5c. This must be wrong because no material object can go faster than the speed of light.

(4)

• Thus for v → c, objects become infinitely massive. This means that an infinite amount of energy is required to accelerate an object to the speed of light.

m0 1 − v 2 /c2

m= p

• Einstein resolved this dilemma by deriving a corrected formula for adding velocities: v +v v = (5) BM M G 1+ 2

• Thus, the speed of light is the ultimate speed for any material object.

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7 – Relativistic Energy

8 – Summary

• Besides momentum, also the definition of kinetic energy must be modified in relativistic mechanics. The formula KE = mv 2 /2 is replaced by KE = mc2 − m0 c2 • The term E0 = m0 c2 is called the rest energy of the object. • The total energy is given by E = mc2 = KE + m0 c2 or m0 c

2

1 − v 2 /c2

(6)

Special Relativity

Concept

valid for v c

valid for all speeds

Time Interval, ∆t =

∆t0

∆t0 γ

Length, L =

L0

L0 /γ

Mass, m =

m0

m0 γ

Momentum, p =

m0 v

Kinetic Energy, KE =

1 2 2 m0 v

Total Energy, E =

KE + PE

m0 vγ mc − m0 c2 mc2

=

( AC + CB ) (1+ AC AB 2 )

v

+v

2

=v

v

Addition of velocities

In words: mass and energy are equivalent

• Relation between relativistic total energy, E, and momentum p:

γ=p

E 2 = p2 c2 + (m0 c2 )2 Spring 2005

Classical Physics

E = γm0 c2 = p

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= 0.96c in the above example.

• Applying this formula gives v

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1 ; E 2 = p2 c2 + (m0 c2 )2 2 2 1 − v /c Spring 2005

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Special Relativity and Quantum Physics • The theoretical predictions for the distribution of wavelengths for thermal radiation of a black body did not match the experimental data (black body: an ideal system that absorbs all radiation incident on it):

9 – Blackbody Radiation and Planck’s Hypothesis Between 1900 and 1930 a revolution took place in physics. The field of Quantum mechanics was developed. This approach is highly successful in explaining the behavior of atoms.

Theory predicted that at short wavelengths the amount of radiation should increase, which is in clear contradiction with data.

• It all started with blackbody radiation. An object at any temperature is known to emit electromagnetic radiation (so-called thermal radiation). The characteristics of this radiation depend on the temperature and the properties of the object.

• In 1900, Planck developed a formula for black body radiation which was in complete agreement with data at all wavelengths. Planck’s model was based on the idea that the walls of a black body radiator were composed of resonators which could only have certain discrete amounts of energy E , given by

• Thermal radiation consists of a continuous distribution of wavelengths from the infrared to ultraviolet portions of the spectrum. • Thermal radiation was thought to originate from accelerated charged particles near the surface of an object which emit radiation much like small antennas. Dr.D.Wackeroth

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E = nhf PHY102A

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• In the outgoing 19th century, experiments showed that when light is incident on certain metallic surfaces, electrons are emitted from these surfaces. This phenomenon is called the photoelectric effect, and the emitted electrons are called photoelectrons.

(8)

When monochromatic light shines onto a metal plate, a current is detected in the ammeter A. For increasing voltage V between the metal plate and the collector, the current reaches a maximum: all photoelectrons are collected on A.

(9)

A molecule or atom will radiate or absorb energy only when it changes quantum states. • The key point in Planck’s theory is the assumption of quantized energy states, which was pretty radical.

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10 – The Photoelectric Effect

• The resonators emit energy in discrete units called quanta or photons by “jumping” from one quantum state to another. The energy of a photon is given by E = hf

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where n is a positive integer (n = 1, 2, 3, . . .) called a quantum number and f is the frequency of vibration of the resonators. The factor h is a constant known as Planck’s constant, given by h = 6.626 × 10−34 J · s

(7)

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• For a negative potential −V0 , no photoelectrons will reach A. This stopping potential is independent from the radiation intensity. The maximum kinetic energy of the photoelectrons is related to V0 by

KE

= eV0

• In Einstein’s theory light is a stream of photons. Each photon has an energy E = hf

(10)

(11)

• When the photon’s energy is transferred to an electron in the metal, the energy acquired by the electron is E = hf .

• In 1905, Einstein developed a theory which explained the photoelectric effect. The major features of his theory are:

1. No electrons are emitted if the frequency of the light is below a threshold frequency f . This is inconsistent with the wave theory of light.

• A certain energy W0 (so-called work function) is required to free electrons from the metal surface. The maximum kinetic energy of the ejected photoelectrons is then given by

2. The number of photoelectrons is proportional to the light intensity.

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KE

3. The maximum kinetic energy of photoelectrons is independent of the light intensity but increases with the light frequency.

= hf − W0

(12)

which is in agreement with the observation.

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Special Relativity and Quantum Physics • To explain the effect, Compton assumed that if a photon behaves like a particle, its collision with other particles is similar to that between billiard balls.

11 – Compton Scattering • Further justification for the photon theory of light came from an experiment conducted by Compton in 1923.

• Hence energy and momentum are conserved. When a photon collides with an electron which is at rest, part of the photon energy and momentum are transfered to the electron → the energy of the scattered photon are lowered and the wavelength increases.

• He directed an X-ray beam of wavelength λ toward a block of graphite. The scattered X-rays had a longer wavelength, λ0 > λ. The shift in wavelength depended on the angle at which the X-rays were scattered.

• The shift in wavelength is given by ∆λ = λ0 − λ =

h (1 − cos θ) m0 c

(13)

where m0 is the rest mass of the electron and θ is the angle between the directions of the scattered and incident photons. Dr.D.Wackeroth

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Special Relativity and Quantum Physics 12 – The Wave Properties of Particles • In 1924, de Broglie proposed that all forms of matter have wave as well as particle properties.

• How can light be considered a photon when it exhibits wave-like properties? Is light a wave or a particle?

• For a photon, the energy and momentum are given by (m0 = 0)

• The answer depends on the specific phenomenon observed.

E = hf = Light has a dual nature; it exhibits both wave and photon characteristics.

hc E h ; p= = λ c λ

(14)

• de Broglie suggested that

If light has a dual nature, how about massive particles ?

Material particles of momentum p should also have wave properties and a corresponding wavelength.

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h h = p mv

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Special Relativity and Quantum Physics 13 – The Uncertainty Principle

• Because the momentum of a particle is p = mv, the de Broglie wavelength of a particle is λ=

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• In classical mechanics, it is in principle possible to perform experiments with arbitrary precision. Quantum theory, in contrast, predicts that

(15)

It is fundamentally impossible to make The effects of this wavelength are only observable for particles with very small mass (electrons, protons).

simultaneous measurements of a particle’s position and velocity with infinite precision.

• In 1927, Davisson and Germer discovered that electrons can be diffracted and, thus, behave like waves in certain experiments.

This statement, the so-called uncertainty principle was derived by Werner Heisenberg in 1927.

• Consider a particle moving along the x axis and suppose that ∆x and ∆p represent the uncertainty in the measured position and momentum, respectively. The uncertainty principle requires that

∆x∆p ≥ Dr.D.Wackeroth

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h 4π

(16)

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• If ∆x is made small, ∆p will be large and vice versa. • Another form of the uncertainty principle that applies to the simultaneous measurement of energy and time is ∆E∆t ≥

h 4π

(17)

where ∆E is the uncertainty in a measurement of the energy and ∆t is the time it takes to make the measurement. • Significance of the uncertainty principle: If an experiment is designed to reveal the particle character of, say, an electron, its wave character becomes fuzzy, and vice versa.

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