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General Physics II

Quantum Physics

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Matter Waves• Light behaves like a wave under

some circumstances (interference and diffraction) and like a particle in others (photoelectric effect).

• Louie de Broglie (pronounced day-broy) proposed that matter also has this dual property. de Broglie’shypothesis was verified when it was experimentally shown that electrons (which typically behave like particles of matter) undergo diffraction, which is a wave-like property.

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Matter Waves• de Broglie also obtained the following relation for the

wavelength of a matter wave:

where h is Planck’s constant and p = mv is the momentum of the particle.

• We do not observe wave-like behavior for everyday moving objects such as baseballs because their de Brogliewavelengths are exceedingly small!

• Group Activity: Calculate the de Broglie wavelength of a baseball of mass 0.15 kg and speed 20 m/s.

, (de Broglie wavelength)h hp mvλ= =

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Matter Waves• If the de Broglie wavelength of

an electron beam is comparable to the slit widths and slit spacing of a set of double slits, one obtains the well-known double-slit interference pattern.

Double-slit interferencepattern with electrons

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What Exactly is Wavy About Electrons?• Quantum mechanics (and

experiment) tells us that we cannot predict where any one electron will land in the interference pattern. However, we can calculate exactly its probability of landing in a certain spot.

• It is the probability that the electron will land in the various possible positions in the pattern that exhibits wave-like properties. Bright fringes – higher probability; dark fringes –lower probability.

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Workbook: Chapter 28 Questions 12 (b), (d); 14

Application of Matter Waves: Electron Microscopes

Because of diffraction, the smallest object that can be seenwith an optical microscope has a size of about one wavelength. Thus, you could see smaller objects if the wavelength of the illumination were smaller. Electrons withthe appropriate speed have a much smaller wavelength than visible light. Hence much smaller details can be seenwith a microscope that uses electrons rather than visible light to illuminate a specimen.

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Quantization of Energy• In the classical physics of

motion we have studied thus far, there was no restriction on the energy that a particle could possess. However, the wave nature of particles does, under certain conditions, impose restrictions on the values of energy a particle can have.

• To understand this, consider a standing wave on a string fixed at both ends. The standing wave is created by interference between reflected waves traveling in opposite directions.

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Standing Waves• Only when a whole number of

half-wavelengths fit precisely within the length of the string will there be standing waves. Thus, only certain discrete wavelengths are allowed.

• Precisely the same thing happens when a particle (matter wave) is trapped in a box. The particle moves back and forth, creating a standing matter wave. Only certain de Brogliewavelengths are allowed and therefore only certain values of momentum are allowed (p = h/λ).

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Energy in Quantized• Since there are only certain

allowed values of momentum, there are also only certain allowed values of kinetic energy and hence total energy (there is no potential energy inside the box because the particle does not interact with anything except when it hits a wall).

• One finds that for a particle trapped in a one-dimensional box of length L,

• As seen above, the energy is discrete or quantized. n is called a quantum number.

222, = 1, 2, 3,

8nhE n nmL

=

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Workbook: Chapter 28, Question 15

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Energy Levels and Quantum Jumps• The allowed energies, or energy

levels for a particle in a box are shown to the right. The lowest energy level corresponds to n = 1. This is the ground state. In the absence of external interactions, the particle will be in the ground state (which does not have zero energy!).

• In order to make a transition or quantum jump to a higher level, the particle must absorb energy precisely equal to the energy difference between the two levels.

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Energy Levels and Quantum Jumps• For example, if the particle is in

the ground state (n = 1) and a photon of energy Eph = E3 – E1 is absorbed, the particle will make a quantum jump to the n = 3 level. Since Eph = hf, the frequency of the required photon is given by

• When a system is in a higher energy state than the ground state, it is said to be excited. A system in an excited state will quickly jump down to lower energy levels until it returns to the ground state. With each jump, the excess energy is emitted as a photon.

3 1( )/ .f E E h= −

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Energy Levels and Quantum Jumps• The photon emitted in each

quantum jump has an energy given by Eph = Einitial – Efinal and a frequency given by

(Emitted photon frequencies)• Note that these frequencies will

also have discrete values and so the light emitted by an excited quantum system will be a discrete spectrum, unlike the rainbow spectrum, which is continuous, i.e., all possible frequencies are allowed between red and violet.

( )/ .initial finalf E E h= −

The wavelength of the emitted photon is given by

.phcfλ =

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Textbook: Chapter 28, Problems 39, 41

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Heisenberg Uncertainty Principle• Consider a particle (matter

wave) that undergoes single-slit diffraction. Note that as the slit is made narrower the diffraction pattern gets wider. Thus, as our knowledge of the x-component of the position of the particle gets better (narrower slit), our knowledge of the x-component of its velocity (and momentum) gets worse. (The x-component of the velocity determines where in the pattern the particle ends up. There is a larger range of possible values with a wider pattern.)

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Heisenberg Uncertainty Principle• Thus, we cannot simultaneously minimize the uncertainties in

both x-component of position and the x-component of momentum. This is essence of the Heisenberg Uncertainty Principle. The more precise a measurement of the position of a particle is, the less precise a measurement of its momentum will be (and vice-versa).

• Note that these uncertainties are only significant for small quantum particles like electrons and atoms. They are completely negligible for measurements involving non-quantum objects such as baseballs and model rockets.

• The Uncertainty Principle is a direct result of the wave-like properties of material particles. Waves are intrinsically “spread out” and so their positions are inherently uncertain.