E.2 Quantum Physics

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2024/07/05 08:52

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E.2.1 Photoelectric Effect:

E.2.1-1 Photoelectric effect from the metal surface

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Photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material.

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Electrons emitted in this manner are called photoelectrons.

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This effect was experimentally found by Rober Einstein, the results of the experiment disagree with classical electromagnetism, which predicts that continuous light waves transfer energy to electrons, which would then be emitted when they accumulate enough energy.

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The experiments instead show that the electrons are emitted only when the light has enough frequency, regardless of the light’s intensity or duration of exposure.

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Einstein came along with a new proposal - Wave-particle duality.

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That the beam of light in this is not a wave propagating through space but is made up of discrete energy packets, the photon particle.

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His detailed explanation was:

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Electrons at the material’s surface need a certain minimum energy in order to escape from the surface. This minimum energy is called the work function of the metal given the symbol .

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The UV light energy arrives in lots of little packets of energy- the packets are called photons.

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The energy in each packet is fixed by the frequency of UV light that is being used, whereas the number of packets arriving per second is fixed by the intensity of the source.

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The energy carried by a photon is given by the equation:

E=hf

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Where E is energy in joules, h is Plank’s constant, and f is the frequency of light in Hz.

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Any extra energy would be retained by the electron as kinetic energy.

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Above the threshold frequency, incoming energy of photons = work function + kinetic energy

E_{\text{max}} = hf - \phi

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Stopping potential is the minimum negative voltage applied to the anode to stop the photocurrent.

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The maximum kinetic energy of the electrons equals the stopping voltage.

E.2.1-2 Relationship between the stopping potential and frequency

\frac{1}{2} m v^2 = V_s e

E.2.2 Atomic Spectra and Atomic Energy States

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As discussed in topic 7, atomic spectra (both emission and absorption) provide evidence for the quantization of the electron energy level.

Hydrogen Spectrum

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We can mathematically describe this quantization of the electron energy level for a hydrogen atom, which is called the Rydberg formula:

\frac{1}{\lambda} = R_H \left( \frac{1}{n^2} - \frac{1}{m^2} \right)

\lambda

= the wavelength of electromagnetic radiation emitted in a vacuum, RH = the Rydberg constant for hydrogen, n = the principal quantum number of an energy level, m = the principal quantum number of an energy level for the atomic electron transition

E.2.3 Schrodinger Model of the Atom

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Schrodinger has built on the concept of matter waves and proposed an alternative model of the hydrogen atom using wave mechanics.

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The Copenhagen interpretation is a way to give a physical meaning to the mathematics of wave mechanics.

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To describe this model, the Schrodinger equation must be used.

Schrodinger Equation Derivation

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Notice De Broglie equation λ=hp\lambda = \frac{h}{p}λ=ph​ and energy equation E=hcλE = \frac{hc}{\lambda}E=λhc​

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The description of particles in quantum mechanics is in terms of the wave function Ψ\PsiΨ, which is a complex multi-variable function that relates both space and time.

Mathematically:

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ψ(x,t)=Acos⁡(kx−ωt)+iAsin⁡(kx−ωt)\psi (x,t) = A \cos(kx - \omega t) + i A \sin(kx - \omega t)ψ(x,t)=Acos(kx−ωt)+iAsin(kx−ωt)

From Euler Identity, we can rearrange the wave function into a polar form:

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ψ(x,t)=Aei(kx−ωt)\psi (x,t) = A e^{i(kx - \omega t)}ψ(x,t)=Aei(kx−ωt)

Now, take the partial derivative with respect to time and position respectively,

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∂∂xψ(x,t)=ikAei(kx−ωt)\frac{\partial}{\partial x} \psi(x,t) = i k A e^{i(kx - \omega t)}∂x∂​ψ(x,t)=ikAei(kx−ωt)   ∂∂tψ(x,t)=−iωAei(kx−ωt)\frac{\partial}{\partial t} \psi(x,t) = - i \omega A e^{i(kx - \omega t)}∂t∂​ψ(x,t)=−iωAei(kx−ωt)

Multiply a factor

i \frac{h}{2\pi}

(which was stated by Schrodinger)

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ih2π∂∂xψ(x,t)=−i2h2πkAei(kx−ωt)=pψ(x,t)i \frac{h}{2\pi} \frac{\partial}{\partial x} \psi(x,t) = - i^2 \frac{h}{2\pi} k A e^{i(kx - \omega t)} = p \psi(x,t)i2πh​∂x∂​ψ(x,t)=−i22πh​kAei(kx−ωt)=pψ(x,t)

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ih2π∂∂tψ(x,t)=i2h2πωAei(kx−ωt)=−Eψ(x,t)i \frac{h}{2\pi} \frac{\partial}{\partial t} \psi(x,t) = i^2 \frac{h}{2\pi} \omega A e^{i(kx - \omega t)} = -E \psi(x,t)i2πh​∂t∂​ψ(x,t)=i22πh​ωAei(kx−ωt)=−Eψ(x,t)

Now denote

\frac{h}{2\pi} = h'

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pψ(x,t)=−ih′∂∂xψ(x,t)p\psi(x,t) = - ih' \frac{\partial}{\partial x} \psi(x,t)pψ(x,t)=−ih′∂x∂​ψ(x,t)

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Eψ(x,t)=ih′∂∂tψ(x,t)E\psi(x,t) = ih' \frac{\partial}{\partial t} \psi(x,t)Eψ(x,t)=ih′∂t∂​ψ(x,t)

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p=−ih′∂∂x(differentiation operator)p = - ih' \frac{\partial}{\partial x} \quad \text{(differentiation operator)}p=−ih′∂x∂​(differentiation operator)

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E=ih′∂∂t(differentiation operator)E = ih' \frac{\partial}{\partial t} \quad \text{(differentiation operator)}E=ih′∂t∂​(differentiation operator)

Since

Ek = \frac{p^2}{2m}, \quad E = Ek + Ep

we can deduce that

ih' \frac{\partial}{\partial t} = \frac{1}{2m} \left( -ih' \frac{\partial}{\partial x} \right)^2 + U

Multiply the wave function on both sides

ih' \frac{\partial}{\partial t} \psi(x,t) = -\frac{h'^2}{2m} \nabla^2 \psi(x,t) + U \psi(x,t)

\nabla =

the gradient, the differentiation operator

Which gives us the one-dimensional time-dependent Schrodinger equation.

For 3-d, the equation becomes

i h' \psi(x,y,z,t) = -\frac{h'^2}{2m} \nabla^2 \psi(x,y,z,t) + U \psi(x,y,z,t)

E.2.4 Pair Production and Pair Annihilation

Pair Production

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Where the electric field is strong near the atomic nucleus, a photon of the right energy can turn into a particle along with its antiparticle.

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The examples are:

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Electron and positron

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Proton and antiproton

E.2.4-1 Feynmann diagram of electron pair production

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The mass is conserved between the particles, which means that the photon must have enough energy to create the masses of the two particles.

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The minimum energy required is given in the equation:

E = 2mc^2

m = the rest mass of the particle/antiparticle, c = the speed of electromagnetic waves in a vacuum

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Some characteristics of electron pair production:

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The gamma-ray energy must have at least 1.02 MeV

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Any photon energy in excess is converted into the kinetic energy

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Occurs in the vicinity of an orbital electron (more energy will be needed)

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Positron and electron spiraling in opposite directions in the applied magnetic field

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Recoiling electrons gain kinetic energy

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The threshold energy needed is 4mc2

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The equation for the electron pair production is:

\gamma + e^- \rightarrow e^- + e^- + e^+

E.2.4-2 Bubble champer tracks of electron pair production

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Annihilation happens when a particle collides with its antiparticle, then they form two photons.

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The total energy of the photons is equal to the total mass-energy of the annihilating particles.

The Heisenberg uncertainty principle is defined as:

\Delta x \Delta p \geq \frac{h}{4\pi}

Δx

= the change in position,

Δp

= the change in momentum.

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This equation states that:

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If ΔxΔxΔx is small, ΔpΔpΔp must be large

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The exact values of xxx and p cannot be determined at the same time

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This is because, for the extremely large momentum, any radiation of the wavelength would impact energy to the nucleus, which will make the position immeasurable.

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Can be used for some other conjugate variables:

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xxx and ppp

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EEE and t

Another uncertainty principle based on pair production is:

\Delta x \Delta p \geq \frac{h}{4\pi}

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The Implication of quantum mechanics is that it is not deterministic, and we cannot predict the results of a single experiment.

Quantum Tunneling

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A particle that is trapped because its energy E is less than the potential energy for escape is trapped by the potential barrier.

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From a quantum physics perspective, since the particle’s wave function is continuous and does not drop to zero, the amplitude of the probability decreases exponentially and continues outside the barrier.

E.2.4-3 Wavefunction change during the quantum tunneling

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There are two examples:

Alpha decay

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The energy is released when an alpha particle inside the nucleus is emitted.

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This example will be examined deeper in topic 12.2.

Tunneling electron microscope

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A very fine metal tip is scanned close to, but not touching the metal surface.

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The electrons on the surface do not have enough potential energy to escape, but some can as a measurable tunneling current.

E.2.4-4 Tunneling electron microscope

E.2.5 Bohr Model of Atom

Bohr postulated that:

An electron does not radiate energy when in a stable orbit.

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The only stable orbits possible for the electron are ones where the angular momentum of the orbits is an integer multiple of h2π\frac{h}{2\pi}2πh​.

Mathematically:

mvr = \frac{nh}{2\pi}

When electrons move between stable orbits, they radiate energy:

F = \frac{mv^2}{r}\\ \frac{e^2}{4\pi\varepsilon_0 r^2} = \frac{mv^2}{r}

From 1st postulate, we can deduce that

v = \frac{nh}{2\pi m r}

Substitute back into centripetal force equation

r = \frac{\varepsilon_0 n^2 h^2}{\pi m e^2}

The total energy of an electron could be calculated:

E = Ek + Ep\\ E = -\frac{1}{2} \frac{e^2}{4\pi\varepsilon_0 r} = -\frac{m e^4}{8 \varepsilon_0^2 n^2 h^2}

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This equation has shown that:

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The electron is bound to the proton because overall it has negative energy.

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The energy of an orbit is proportional to 1n2\frac{1}{n^2}n21​(all the other values are constant), this approximately gives us En=−13.6n2E_n = -\frac{13.6}{n^2}En​=−n213.6​

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Moreover, the second postulate can be used to predict the wavelength of radiation emitted when an electron makes a transition between stable orbits.

E = hf = E_2 - E_1\\ E = \frac{me^4}{8\varepsilon_0^2 h^2} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

However, notice

f = \frac{c}{\lambda}

Thus,

\frac{1}{\lambda} = \frac{me^4}{8 \varepsilon_0^2 c h^3} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)

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This equation is of the same form as the Rydberg equation, the values predicted by this equation are in very good agreement with experimental measurement.

Some of the Limitations of Bohr’s Model

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If the same approach is used to predict the emission spectra of other elements, it fails to predict the correct values for atoms with more than one electron.

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The first postulate about angular momentum has no theoretical justification

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Theory predicts that electrons should not be stable in circular orbits around a nucleus. Any accelerated electron should radiate energy, an electron in a circular orbit is accelerating so it should radiate energy and thus spiral into the nucleus.