Introduction to Quantum Mechanics: Matter Waves and Interference, Exams of Quantum Physics

Download Introduction to Quantum Mechanics: Matter Waves and Interference and more Exams Quantum Physics in PDF only on Docsity! “We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery.” --Richard P. Feynman Introduction to Quantum Mechanics • This week and the next are critical for the course • Week 3 – Lect. 5,6: Light as Particles -- Particles as waves -- Wave-Particle Duality -- Probability --Uncertainty Principle -- “Weirdness” of Quantum Mechanics • Week 4 – Lect. 7,8: Schrödinger Equation -- Definite predictions of quantum behavior -- Examples of particles in infinite wells, finite wells -- Leads up to rest of course • Midterm Exam – Monday, Feb. 16 - will cover Lectures 1-7 and qualitative aspects of lecture 8 • Week 4 – The discussion and the quiz will be on material in lecture 5-6 • Week 4 – Online homework covers material in lecture 7 – due on Thur. Feb. 19, but we strongly encourage you to look at the homework before the midterm! •Review – Sunday, Feb. 15 - office hours Feb. 15 and 16 Interference – What really counts Before, we discussed interference from a classical perspective. The amplitudes from two or more physical paths interfered- if nothing else (like polarization) distinguished the two parts. A more modern perspective is that indistinguishable processes interfere. Example: ψupper is the amplitude corresponding to the process by which a photon leaves the source, travels through the upper slit, and reaches the point y on the screen. ψlower is the amplitude corresponding to the process by which a photon leaves the source, travels through the lower slit, and reaches the point y on the screen. If these processes are somehow distinguishable (i.e., there’s some evidence left somewhere about which process occurred), then we add the probabilities together: P(y) = P(y, by way of upper slit) + P(y, by way of lower slit) = |ψupper|2 + |ψlower|2 If these processes are in principle indistinguishable, add the amplitudes together and take the absolute square to get the probability: P(y) = |ψupper + ψlower|2 Now let’s modify the experiment a bit. (Recall that EM waves can be polarized – electric field in the vertical or horizontal directions)  Send in unpolarized photons  Cover the upper slit with a vertical polarizer and cover the lower slit with a horizontal polarizer Now the resulting pattern will be: Act 1 |ψ1 + ψ2|2A) B) H ?? Unpolarized V |ψ1|2 + |ψ2|2 Act 1 - Solution  In this case, the photon’s polarization serves to label which way it went; because the two processes are in principle distinguishable there is no interference.  Note, that we don’t need to actually measure the polarization. It is the mere fact that one could measure it that destroys the interference. Bonus Question: How could we recover the interference? Now let’s modify the experiment a bit. (Recall that EM waves can be polarized – electric field in the vertical or horizontal directions)  Send in unpolarized photons  Cover the upper slit with a vertical polarizer and cover the lower slit with a horizontal polarizer Now the resulting pattern will be: |ψ1 + ψ2|2A) B) H ?? Unpolarized V |ψ1|2 + |ψ2|2 electron gun Ni Crystal detector θ  In 1927-8, Davisson & Germer* showed that, like x-rays, ELECTRONS can also diffract off crystals ! I( θ) θ Interference peak ! 0 60o Electrons can act like waves! Just like photons! What does this mean? In discussion section: θ Matter Waves  Demonstration that particles (electrons) act like waves: Interference! *Work done at Bell Labs, Nobel Prize Act 2: Matter wavelengths  What size wavelengths are we talking about? Consider a photon with energy 3 eV, and therefore momentum p = 3 eV/c. Its wavelength is: a) λe = λp b) λe < λp c) λe > λp  What is the wavelength of an electron with the same momentum?  What size wavelengths are we talking about? Consider a photon with energy 3 eV, and therefore momentum p = 3 eV/c. Its wavelength is: a) λe = λp b) λe < λp c) λe > λp  What is the wavelength of an electron with the same momentum? λe = h/pe Same relation forparticles and photons. Compared to the energy of the photon (given above):  Note that the kinetic energy of the electron is different from the energy of the photon with the same momentum (and wavelength): Act 2: Matter wavelengths - Solution Interference of larger particles  Matter-wave interference has now been demonstrated with electrons, neutrons, atoms, small molecules, BIG molecules, & biological molecules  Recent Example: Interference of C60, a.k.a. “fullerenes”, “buckyballs” [A. Zeilinger (U. Vienna), 1999] Mass = (60 C)(12 g/mole) = 1.2 x 10-24 kg λ = h/p = 2.9 pm (c.f. C60 is ~ 1 nm across!) FYI: More on Interference of larger particles  Using a velocity selector, they could make the atoms more monochromatic  improved interference:  In 2003 interference was observed with porphyrin, a bio. molecule: Original distribution Narrowed distribution Now they’re trying to do something like this with a virus! a) Calculate the wavelength of an electron that has been accelerated from rest across a 3-Volt potential difference (me = 9.11×10-31 kg). [0.71 nm] b) Do the same for a proton (mp = 1.67×10-27 kg). [17 pm] c) Calculate the wavelength of a major league fastball (mbaseball = 0.15 kg, v = 50 m/s). [8.8 x 10-35 m] a) E = e V = 4.8×10-19 J p = √(2meE) = 9.35×10-25 kg m/s λ = h/p = 7.1×10-10 m = 0.71 nm b) p = √(2mpE) = 4.00×10-23 kg m/s λ = h/p = 1.7×10-11 m c) p = mv = 7.5 kg m/s λ = h/p = 8.8×10-35 m This is from Physics 212. This is Physics 211. This is Physics 214. E is the same, because the electric charge is the same. Mass is bigger ⇒ λ is smaller. SI units were designed to be convenient for macroscopic objects. Example Problem: Wavelengths of various “particles” Solution: Quantum mechanical wave effects are negligible in the motion of macroscopic objects. The wavelength is many orders of magnitude smaller than any distance that has ever been measured. Application of Matter Waves: Electron Microscopy Scientists and engineers - such as those here at the Materials Research Lab and the Beckman Institute - use “electron microscopy” to study nanometer-scale structures in materials and biological systems (//www.itg.uiuc.edu/) Compound eye of a flyCu-In alloy  Electron Microscopy of a Virus: You wish to observe a virus with a diameter of 20 nm, which is much too small to observe with an optical microscope. Calculate the voltage required to produce an electron DeBroglie wavelength suitable for studying this virus with a resolution of dmin = 2 nm. The “f-number” for an electron microscope is quite large: f/D ≈ 100. (Hint: First find λ required to achieve dmin with the given f/D. Then find E of an electron from λ.) Example: Imaging a Virus* Answer: 5.6 kV object Electron optics D f electron gun Solution To accelerate an electron to an energy of 5.6 keV requires 5.6 kilovolts . (The beauty of electron-volt units)  Electron Microscopy of a Virus: You wish to observe a virus with a diameter of 20 nm, which is much too small to observe with an optical microscope. Calculate the voltage required to produce an electron DeBroglie wavelength suitable for studying this virus with a resolution of dmin = 2 nm. The “f-number” for an electron microscope is quite large: f/D ≈ 100. (Hint: First find λ required to achieve dmin with the given f/D. Then find E of an electron from λ.) object Electron optics D f electron gun Heisenberg Uncertainty Principle  So, particles (electrons, photons, etc.) also have wave-like properties – reflects a fundamental uncertainty in the “knowability” (existence?) of the particle’s precise location.  For classical waves one can produce a localized “wave packet” by superposing waves with a range of wave vectors Δk. E.g.:  From Fourier analysis: Δk·Δx ≥ 1. (with a little license on meaning of “Δ”)  Interpretation: To make a short wavepacket requires a broad spread in wavelengths. Conversely, a single- wavelength wave would extend forever.  From the quantum relation between momentum and wave vector (p = h/λ = (h/2π)/(λ/2π) = ħk), there is always a trade-off between the spread in the a particle’s likely locations x and likely momenta p ħ (Δk·Δx ≥ 1) ⇒ (ħΔk)·Δx ≥ ħ ⇒ Δpx·Δx ≥ ħ This relation is known as the Heisenberg Uncertainty Principle. Δx Why ‘Uncertainty’?  Classically you can’t have  both very small Δk and very small Δx Or  both very small Δf and very small Δt  So why didn’t we call that ‘classical uncertainty’ and make a big fuss about it? Each time you look, you find a local blip that is in a different place (in fact, it is your looking that causes the wavefunction to “collapse”!). If you look many times, you will find a probability distribution that is spread out. But you’re uncertain about where that local blip will be in any one of the times you look -- it could be anywhere in the spread. Because the classical waves were certainly spread out in k and x, and in f and t, and would always show that spread whenever you looked.  The difference for the quantum case is that when you go to look, you can find a local blip that does not show the whole spread. Uncertainty Principle: Diffraction  Look at a familiar example  single-slit diffraction.  The particle’s transverse location is constrained by the slit width a: Δx ≈ a/2  This implies an uncertainty Δpx in the transverse momentum: Δpx ≈ ħ/Δx ≈ 2ħ/a  Geometry relates Δpx to the total momentum p: Δpx = p sinθ = (h/λ) sinθ  Equating these, we find a sinθ ≈ λ/π .  i.e., slightly inside the location of first diffraction minimum: a sinθ = λ Confining the particle’s location transversely leads to a bigger spread in the transverse momentum. a Δpxθ p Uncertainty Principle –Implications  The uncertainty principle explains why (negative) electrons in atoms don’t simply fall into the (positive) nucleus: If the electron were “confined” too close to the nucleus (small Δx), it would have a large Δp, and therefore a very large average kinetic energy (≈ (Δp)2/2m).  The uncertainty principle does not say “everything is uncertain”. Rather, it tells us very exactly where the limits of uncertainty lie when we make measurements of quantum systems.  It appears that features like ‘paths’ do not in general exist, but only exist to a degree determined by not just a single particle but also its environment. One interpretation, then, is that such concepts as orbits of electrons do not exist in nature unless and until we observe them. (Other features can exist precisely.)  The serious evidence involves something called “Bell’s Inequality”.  Serious philosophical issues remain open to vigorous debate.  E.g. whether all outcomes or only one outcome actually occur. Pa eo) WA ‘> a — s : : g S 6UT BOB. Supplementary Problem  A hydrogen atom is about 0.1 nm in diameter. Suppose we wanted to measure the position of its electron with an accuracy of, say, 0.01 nm by scattering a photon off it. How much energy would be transferred to the electron if the photon lost most of its energy in the scattering?  The photon would have to have a wavelength of about 0.01 nm.  The energy of the photon is Since the binding energy of the electron is only about 10 eV, this particular measurement doesn’t exactly leave the atom unscathed.