Physics 2 - Lecture 8: Fundamental of Quantum Mechanics - Pham Tan Thi

Nội dung text: Physics 2 - Lecture 8: Fundamental of Quantum Mechanics - Pham Tan Thi

1. Fundamental of Quantum Mechanics Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Sciences Ho Chi Minh University of Technology
2. Diffraction of Light/Electron by One Slit
3. Double-slits Experiments for Electrons (Wave-like character of electrons)
4. Wave-Particle Duality Energy in Energy out A beam of light can be thought as a flux of particle an electromagnetic wave (Newton/Planck/Einstein) (Huygens/Maxwell/Hertz) Wavelength λ Electric Field Zero mass, speed: c = 3 x 108 m/s Energy carried by each particle: Direction of propagation h = 6.6262 x 10-34 J.s (Planck) c = λν ν (frequency) = 1/T (period) Dispersion relation (Free space) Particle (Photon): Wave (Electromagnetism): - Photoelectric Effect - Interference - Compton Effect - Diffraction
5. For a Non-relativistic Free Particle Momentum is p = mv, here is v is the speed of the particle Total energy of a free particle, E, is kinetic energy: h h h B = = = p mv p2mE p2 mv2 E = K = = 2m 2 Bullet: m = 0.1 kg; v = 1000 m/s 36 ➔ 6.63 10 m B ⇡ ⇥
6. What is the de Broglie’s Wavelength of an Electron? What is the de Broglie’s wavelength of an electron moving at 2.2 x 106 m/s Now this is really really fast: 2.2 million meters per second. But it’s not relativistic. It’s is still slow compared to the speed of light so we can still do everything fairly classically. 31 6 24 p = mv =9.1 10 2.2 10 [kg m/s] = 2 10 N.s ⇥ ⇥ ⇥ ⇥ 34 24 10 = h/p =6.626 10 /2 10 =3.33 10 m ⇥ ⇥ ⇥ This is important number. This speed is a kind of average speed of an electron in ground state of hydrogen.
7. AcceleratedWhat is the deCharges Broglie’s (electrons, Wavelength protons) of a Person?Produce de Broglie’s theory can be extended to show that all matter exhibits the same wave-particle duality as light. This means everything in the universe can act like a wave: h = and h = 6.626 x 10-34 m2kg s-1 mv This shows that an object’s wavelength gets smaller when the more massive it is, and the faster it is moving. If a person has a mass of 75 kg, and is jogging at 8 km/h (which is about 2.2 m/s), then 31 6.626 10 36 = ⇥ =4.016 10 m 7.5 2.2 ⇥ ⇥ This is about 700 billion billion times smaller than the classical electron radius, which is about 2.8 x 10-15 m. Diffraction works best if the slit is about the same size as the wavelength, and so this explains why we do not notice wave-like behavior in human.
8. 38.4 Wave–Particle Duality, Probability, and Uncertainty 1277 h (photon energy in terms E = hƒ = 2pƒ =Uv (38.19b) 2p of angular frequency) Using Eqs. (38.19) in Eq. (38.18), we can rewrite our photon wave equation as (wave function for a Ey x, t = Asin px x - Et U photon with x-momentum (38.20) px and energy E) 1 2 31 2> 4 Since this wave function has a definite value of x-momentum px, there is no uncertainty in the value of this quantity: ¢px = 0 . The Heisenberg uncertainty principle, Eq. (38.17), says that ¢x¢px ÚU2. If ¢px is zero, then ¢x must be infinite. Indeed, the wave described by Eq. (38.20) extends along the entire x-axis and has the same amplitude everywhere. The> price we pay for knowing the pho- ton’s momentum precisely is that we have no idea where the photon is! In practical situations we always have some idea where a photon is. To describe this situation, we need a wave function that is more localized in space. We can create one by superimposing two or more sinusoidal functions. To keep things simple, we’ll consider only waves propagating in the positive x-direction. For example, let’s add together two sinusoidal wave functions like those in Eqs. (38.18) and (38.20), but with slightly different wavelengths and frequencies and hence slightly different values px1 and px2 of x-momentum and slightly different values E1 and E2 of energy. The total wave function is Ey x, t = A1sin p1x x - E1t U + A2sin p2x x - E2 t U (38.21) Consider1 what2 this wave31 function looks2> 4 like at a31 particular instant2> 4 of time, say t = 0. At this instant Eq. (38.21) becomes Ey x, t = 0 = A1sin p1x x U + A2sin p2x x U (38.22) Figure 38.19a 1is a graph2 of the individual1 > 2 wave functions1 > at2 t = 0 for the case A2 =-A1, and Fig. 38.19b graphs the combined wave function Ey x, t = 0 given by Eq. (38.22). We saw something very similar to Fig. 38.19b in our dis- cussion of beats in Section 16.7: When we superimposed two sinusoidal1 waves2 with slightly different frequencies (see Fig. 16.24), the resulting wave exhibited amplitude variations not present in the original waves. In the same way, a photon represented by the wave function in Eq. (38.21) is most likely to be found in the regions where the wave function’s amplitude is greatest. That is, the photon is localized. However, the photon’s momentum no longer has a definite value because we began with two different x-momentum values, px1 and px2. This agrees with the Heisenberg uncertainty principle: By decreasing the uncertainty in the photon’s position, we have increased the uncertainty in its momentum. 38.19 (a) Two sinusoidal waves with slightly different wave numbers k and hence slightly different values of momentum px =Uk shown at one instant of time. (b) The superposition of these waves has a momentum equal to the average of the two individual values of momentum. The amplitude varies, giving the total wave a lumpy character not possessed by either individualSuperposition wave. Principle Ey(x) (a) 0 x Ey(x) (b) 0 x
9. Heisenberg Uncertainty Principle If Δx is measured accurately, i.e., x 0 ⇒ px ! !1 The principle applies to all canonically conjugate pairs of quantities in which measurement of one quantity affects the capacity to measure the other. For examples, Energy E and time t ~ Et 2 and Angular momentum L and angular position θ ~ L✓ 2
10. Determination of the Position of a Particle by a Microscope We can’t measure the momentum of the electron prior to illumination. So there is uncertainty in the measurement of momentum of the electron. The scattered photon can enter the microscope anywhere between the angular range +i/-i. The momentum of the scattered photon is (according to de Broglie) h p = Its x-component can be given as 2h p = sini x
11. Applications of Heisenberg Uncertainty Principle (i) Non-existence of electron in nucleus Order of diameter of an atom ~5 x 10-15 m 15 If electron exists in the nucleus then (x) =5 10 m max ⇥ ~ xp x 2 ~ (x) (p ) = max x min 2 ~ 20 1 (px)min = =1.1 10 kg.m s 2(x)max ⇥ then E = pc = 20 MeV Thus kinetic energy of an electron must be greater than 20 MeV to be a part of nucleus. Experiments show that the electrons emitted by certain unstable nuclei don’t have energy greater than 3-4 MeV. Thus we can conclude that the electrons cannot be present within nuclei.
12. Concept of Bohr Orbit violates Uncertainty Principle According to the concept of Bohr orbit, energy of an electron in an orbit is constant, i.e., ∆E = 0 ~ Et 2 ➔ t !1 All energy states of the atom must have an infinite life-time. But the excited states of the atom have life-time ~10-8 sec. The finite life-time ∆t gives a finite width (uncertainty) to the energy levels.
13. Taylor’s experiment (1908): double slit experiment with very dim light: interference pattern emerged after waiting for few weeks. Interference cannot be due to interaction between photons, i.e. cannot be outcome of destructive or constructive combination of photons. Interference pattern is due to some inherent property of each photon - it “interferes with itself” while passing from source to screen. Photons don’t “slit” —> light detectors always show signals of same intensity Slits open alternately: get two overlapping single-slit diffraction patterns - no two-slit interference Add detector to determine through which slit photon goes: —> no two interference Interference pattern only appears when experiments provides no means of determining through which slit photon passes.
14. Double-slit Exp. — Wave vs Quantum Wave theory: Quantum theory: Patterns of fringes: Patterns of fringes: Intensity bands due to variations Intensity bands due to variations in square of amplitude, A2, of in probability, P, of a photon resultant wave on each point on striking points on screen. screen. Role of the slits: Role of the slits: to provide two coherent sources to present two potential routes by of the secondary waves that which photon can pass from interfere on the screen. source to screen.
15. Properties of Wave Function 1. It must be finite everywhere. If ψ is infinite for a particular point, it means an infinitely large probability of finding the particles at that point. This would violate the uncertainty principle. 2. It must be single valued. If ψ has more than one value at any point, it mean more than one value of probability of finding the particle at that point which is obviously ridiculous. 3. It must be continuous and have a continuous first derivative everywhere. ~2 @2 (x) + U(x) (x)=E (x) 2m @x2
16. Derive the Schrodinger Equation (Cont.) Second differential equation (1) with respect to position: 2 @ 2 2 (kx !t) 2 2 = i k e = i k @t2 p 2⇡ h h Substitute to (1.2): k = where k = , p = , ~ = ~ 2⇡ 2 2 @ 2 2 2 2 @ = i k p = ~ @x2 @x2 p2 p2 E = + U E = + U 2m 2m ~2 @2 ~2 + U = E General Form, 2 + U = E 2m @x2 2mr
17. Schrodinger Equation for Particle in a Box ~2 @2 (x) + U(x) (x)=E (x) 2m @x2 +∞ x ≤ 0 For a particle in a box with U(x) = 0 0 < x < L infinitely hard wall, -∞ x ≥ L ~2 @2 (x) = E (x) 2m @x2 We can re-write it as @2 (x) 2mE = (x) @x2 ~2 @2 (x) 2mE = k2 (x) k2 = @x2 ~2
18. ⇡ k = n (n = 1, 2, 3, ) n L Particle in a Box: Energy Levels: 2 ⇡ 2 ~2k2 ~ n ⇡2~2 h2 E = n = L = n2 = n2 n 2m 2m 2mL2 8mL2 ✓ ◆ ✓ ◆ The allowed wave functions are given by n⇡ (x)=Asin x n L ⇣ ⌘ The normalized wave function: 2 n⇡ (x)= sin x n L L r ⇣ ⌘
19. Example 1: Electron in a 10 nm wide well with infinite barriers. Calculate Eo for L = 10 nm. Example 2: Assume that a photon is absorbed and the electron is transferred from the ground state (n =1) to the second excited state (n = 3). What was the wavelength of the photon?
20. Particle in a box with “Finite Walls” ~2 @2 + U = E U now has a finite value 2m @x2
21. Tunneling Effects