Module 1- Quantum Mechanics-Completed.pdf
- 1. Basic Ideas of Quantum Mechanics M.R. Soleymani Oct. 19, 2020
- 2. Outline u Linear u Measurement and Interference u Uncertainty Principle
- 3. Introduction u Quantum Mechanics is the theory explaining the physical properties of matter at an atomic level. It describes the behaviour of small (elementary) particles. These include electrons, photons, neutrons, protons and any other elementary particle. u While everything in the world including our body is made of these particles, the quantum mechanical behaviour of matter shows itself only when we move to small size matter at the atomic level. u At microscopic level, laws of quantum mechanics pop on and we observe behaviour of the matter in a way completely different from what we expect based on our experience with macroscopic phenomena. u The reason is that large objects consist of a huge number of elementary particles that interact with one another and result in an overall (average) property that is quite stable. u On the small scale, when one observes one or a few elementary particles in isolation from the rest of the world then the “spooky” (according to Einstein) properties of matter shows up.
- 4. Introduction u In this note, we explain some of the basic ideas of Quantum Mechanics useful to quantum computation. u We use a reference Lectures on Physics by Richard Feynman, Vol III: Quantum Mechanics. We only use whatever is needed in a course on quantum computation and quantum communication. Those who like to have more insight into the theory can refer to Feynman lectures or many other references therein. u At microscopic level, laws of quantum mechanics pop on and we observe behaviour of the matter in a way completely different from what we expect based on our experience with macroscopic phenomena. u The reason is that large objects consist of a huge number of elementary particles that interact with one another and result in an overall (average) property that is quite stable. u In the small scale, when one observes one or a few elementary particles in isolation from the rest of the world then the “spooky” (according to Einstein) properties of matter shows up.
- 5. Experiment with Bullets u In order to explain the interference phenomenon, we consider several experiments suggested by Feynman. u The first experiment is done with bullets. There is gun that shoots bullets randomly at different angles. There is a wall in front of it with two slots and behind it a screen with a detector that moves up or down to capture the bullets and register their location. u If we close slot 2 and let bullets go through slot 1 only, the probability of having bullets at different parts of the wall is 𝑃!. u Closing slot 1, we get 𝑃". With bot slots open, we get 𝑃!" = 𝑃! + 𝑃".
- 6. Experiment with Waves u Next, we consider an experiment with water waves: The height of the wave is given by a sinusoidal of the form ℎ!𝑒#$% and ℎ"𝑒#$% for the waves from slots 1 and 2, respectively u The intensity of the wave from sots 1 and 2 is 𝐼! = ℎ! " and 𝐼" = ℎ" " , respectively. u The intensity when both slots are open is 𝐼!" = ℎ! + ℎ" " = ℎ! " + ℎ" " + 2 ℎ! ℎ" cos 𝛿 , where 𝛿 is the phase difference between ℎ! and ℎ". u Therefore, 𝐼!" = 𝐼! + 𝐼" + 2 𝐼!𝐼"cos(𝛿). So, we observe the peaks and lows.
- 7. Experiment with Electrons u No, let’s try an experiment with electrons coming out of an electron gun similar to ones used in Cathode Ray Tubes (CRT) in old TV sets. Here, when slot 2 is closed, we observe the probability curve 𝑃!and when we close slot 1, we get 𝑃". But when both slots are open instead of 𝑃! + 𝑃" observed in the case of bullets, we observe, the interference patter shown in (3). While this interference phenomenon is hard to explain and even harder to believe it has an easy mathematical explanation. Instead of probabilities to be added to give the total probability, the probability amplitudes are added and 𝑃!" = 𝜙! + 𝜙" " ≠ ∅! " + ∅" " = 𝑃! + 𝑃"
- 8. If a Tree Falls in the Forest u Based on what we saw in the previous slide, we may conclude that the electron behaves like wave. But wait! Let’s repeat the experiment with electron but this time put a light source behind the wall. So that we can watch through which slot the electron goes. u With this setup, the interference disappears and probabilities add up. u It seems that electron acts as wave when it is not observed and as particle when observed. u Later, we discuss this in terms of unitary evolution and measurement operation. While measurement destroys the interference and collapses the qubit into a known state, unitary evolution preserves the state by preserving the interference.
- 9. Uncertainty Principle u The disappearance of the interference due to Heisenberg’s uncertainty principle. According to the original version of the Heisenberg’s principle, if we know the momentum 𝑝& of a particle in 𝑥-direction with an accuracy (standard deviation) Δ𝑝&, we cannot know its 𝑥-position more accurately than Δ𝑥 ≥ ℏ/2Δ𝑝&, where ℏ = ℎ/2𝜋 is a constant called the reduced Planck constant equal to 1.05×10'() joule-seconds. u To see the relationship between the special case of uncertainty principle and the disappearance of interference, lets consider the experiment with some modification:
- 10. Uncertainty Principle u In the new setup, the wall is replaced by a panel placed between two rollers so it can move vertically. Assume that the detector is at 𝑥 = 0. Any particle going through the slit 1, will deflect downward to get to the detector. Since the vertical component of the electron momentum is changed, the plate must recoil with an equal momentum in the opposite direction. The plate will get an upward kick, On the other hand if the electron goes through the lower whole the push will be downward. u For any location of detector, the momentum received by the plate will have different values foe electron going through hole 1 and 2. Now assume that we want to find which hole the electron went through by just looking at the plate. In order to do this, we need to measure the momentum 𝑝& accurately. But doing that, according to uncertainty principle, we cannot have 𝑥 with accuracy better than Δ𝑥 ≥ ℏ/2Δ𝑝&. u So, the more accurate we measure momentum (smaller ∆𝑝&) less accurate will be our knowledge of the location of the panel in 𝑥-direction. So, the interference pattern shifts up and down resulting in smearing (averaging) of the interference pattern, thus the interference is lost.
- 11. Basic Principles of Quantum Mechanics u Let’s summarize what we discussed in the previous slide: 1. The probability of a given outcome in an ideal experiment is the square of the absolute value of a complex number ∅: 𝑃 = ∅ !, ∅ = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑃 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 2. When an event has several outcomes, the probability amplitudes add up and there is interference: ∅ = ∅" + ∅! 𝑃 = ∅" + ∅! !, 3. If the experiment is performed in a way that the observer knows which outcome has occurred then the interference is lost and: 𝑃 = 𝑃" + 𝑃!