Particle Nature of Light

h f = h f = KEKEmaxmax + + WWoo

Wave-Particle Duality

λλ==h/mvh/mv

Complete the following statement: According to the de Broglie relation, the wavelength of a "matter" wave is inversely proportional to

(a)Planck's constant.

(b) the frequency of the wave.

(c) the mass of the particle.

(d) the speed of the particle.

(e) the momentum of the particle. X

What happens to the de Broglie wavelength of an electron if its momentum is doubled?

(a) The wavelength decreases by a factor of 4. (b) The wavelength increases by a factor of 2.(c) The wavelength increases by a factor of 4. (d) The wavelength decreases by a factor of 2(e) The wavelength increases by a factor of 3.

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Determine the de Broglie wavelength of a neutron (m = 1.67 × 10–27 kg) that has a speed of 5.0 m/s.(a) 79 nm (b) 395 nm (c) 1975 nm(d) 162 nm (e) 529 nm

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Upon which one of the following parameters does the energy of a photon depend?

(a) mass (c) polarization (e) phase relationships

(b) amplitude (d) frequency

For which one of the following problems did Max Planck make contributions that eventually led to the development of the “quantum”hypothesis?

(a) photoelectric effect (d) the motion of the earth in the ether

(b) uncertainty principle (e) the invariance of the speed of light through vacuum

(c) blackbody radiation curves

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Light is usually thought of as wave-like in nature and electrons as particle-like. In which one of the following activities does light behave as a particle or does an electron behave as a wave?

(a) A Young’s double slit experiment is conducted using blue light.

(b) X-rays are used to examine the crystal structure of sodium chloride.

(c) Water is heated to its boiling point in a microwave oven.

(d) An electron enters a parallel plate capacitor and is deflected downward.

(e) A beam of electrons is diffracted as it passes through a narrow slit. X

Wave FunctionWave Function, , ψψ

The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle

Description of wavesDescription of wavesType of waves Variable physical quantityType of waves Variable physical quantityWater waves Height of the water surfaceSound waves Pressure in the mediumLight waves Electric and magnetic fieldsMatter waves Wave function, Matter waves Wave function, ΨΨ

Ψ, the amplitude of a matter wave, is a function of time and position

Probability DensityProbability Density ΨΨ22

The value of Ψ2 for a particular object at a certain place and time is proportional to the probability of finding the object at that place at that time.

For example: Ψ2 =1: the object is definitely thereΨ2 =0: the object is definitely not thereΨ2 =0.4: there is 40% chance of finding the object there at that time.

Ψ2 starts from Schrodinger’s equation, a differential equation that is central to quantum mechanics

WhyWhy ΨΨ22? ? Why notWhy not ΨΨ??Amplitude of every wave varies from –A to +A to –A to +A and so on (A is the maximum absolute value whatever the wave variable is).A negative probability is meaningless.

ΨΨ22 gives a positive quantity that can be compared with experiments. The key point to the wave function is that the position

of a particle is only expressed as a likelihood or probabilityuntil a measurement is made.

The probability the electron will be found at the particular positionis determined by the wave function illustrated to the right of the aperture.When the electron is detected at A, the wave function instantaneously collapses so that it is zero at B.

Question: A large value of the probability density Ψ2 of an atomic electron at a certain place and time signifies that the electron

(a) is likely to be found there(b) is certain to be found there(c) has a great deal of energy there(d) has a great deal of charge there

Answer: a

Question: A moving body is described by the wave function Ψ at a certain time and place. The value of Ψ2 is proportional to the body’s

a. electric field.b. speedc. energy d. probability of being found

Answer: d

Question:What do physicists enjoy the most at baseball games?

Answer: The “wave”.

The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle

1. If an object has a well-defined position at a certain time, its momentum must have a large uncertainty.

2. If an object has a well-defined momentum at a certain time, its position must have a large uncertainty.

Uncertainty Principle

Momentum and position ΔxΔp ≥ h/4π Energy and time ΔEΔt ≥ h/4π

p=h/λ precisex unknown

Δx better defined(narrower wave packet) Δp less defined(greater spread of λ)

Uncertainty principle ΔxΔp≥h/4π

The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle

In the microscopic world where the wave aspects of matter are very significant, these wave aspects set a fundamental limit to the accuracy of measurementsof position and momentum regardless of how good instruments used are.

The uncertainty principle is the physical law which follows from the wave nature of matter

Example: Compare the de Broglie wavelength of 54-eV electrons with that of a 1500-kg car whose speed is 30 m/s.

Solution:For the 54For the 54--eV electroneV electron:KE=(54eV)(1.6x10-19J/eV)=8.6x10-18 JKE=1/2 mv2, mv=(2mKE)1/2

λ=h/mv=h/(2mKE)1/2= 1.7x10-10 m

The wavelength of the electron is comparable to atomic scales (e.g., Bohr radius=5.29x10-11 m). The wave aspects of matter are very significant.

For the car: For the car: λ=h/mv=6.63x10-34 J•s/(1.5x103)(30m/s)= 1.5x10-38 m

The wavelength is so small compared to the car’s dimension that no wave behavior is to be expected.

Question: The narrower the wave packet of a particle is

a. the shorter its wavelengthb. the more precisely its position can be

established c. the more precisely its momentum can be

establishedd. the more precisely its energy can be

established Answer: b

Question: The wave packet that corresponds to a moving particle

a. has the same size as the particle b. has the same speed as the particlec. has the speed of lightd. consists of x-ray

Answer: b

Question: If Planck’s constant were larger than it is,

a. moving bodies would have shorter wavelengthb. moving bodies would have higher energiesc. moving bodies would have higher momentad. The uncertainty principle would be significant on

a larger scale of size

Answer: d

If Planck’s constant were changed to 660 J•s, what would be the minimum uncertainty in the position of a 120-kg football player running at a speed of 3.5 m/s?

(a) 0.032 m (b) 0.13 m (c) 0.50 m(d) 0.065 m (e) 0.25 m

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