Physics 2 - Lecture 10: Atomic Physics - Pham Tan Thi

Nội dung text: Physics 2 - Lecture 10: Atomic Physics - Pham Tan Thi

1. Atomic Physics Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Sciences Ho Chi Minh University of Technology
2. History of Atomic Model • Proposed an Atomic Theory which states that all atoms are small, hard, indivisible and indestructible particles made of a single material formed into different shapes and sizes. • Aristotle did not support his atomic theory. Democritus (460 BC - 370 BC)
3. History of Atomic Model • In 1803, he proposed an Atomic Theory which states that: • All substances are made of atoms; atoms are small particles that cannot be created, divided or destroyed. • Atoms of the same element are exactly alike, and atoms of different elements are different • Atoms join with other atoms to make new John Dalton substances (1766 - 1844) • He calculated the atomic weights of many various elements • He was a teacher at a very young age • He was color blind
4. History of Atomic Model J. J. Thomson (1856 - 1940) • He proved that an atom can be divided into smaller parts • While experimenting with cathode-ray tubes, discovered corpuscles, which were later called electrons • He stated that the atom is neutral • In 1897, he proposed Plum Pudding Model which states that atoms mostly consist of positively charged material with negatively charged particles (electrons) located throughout the positive material • He won the Nobel Prize
5. History of Atomic Model Niels Bohr (1885 - 1962) • In 1913, he proposed the Bohr Model, which suggests that electrons travel around the nucleus of an atom in orbits for definite paths. Additionally, the electron can jump from a path in one level to a path in another level (depending on their energy) • He won the Nobel Prize • He used to work with Ernest Rutherford
6. History of Atomic Model James Chadwick (1891 - 1974) • In 1932, he realized that the atomic mass of most elements was double the number of protons —> discovery of the neutron • He used to work with Ernest Rutherford • He won the Nobel Prize
7. Schrödinger Equation in Three Dimensions • Electrons in an atom can move in all three dimensions of space. If a particle of mass m moves in the presence of a potential energy function U(x,y,z), the Schrödinger equation for the particle’s wave function ψ(x,y,z) is ~2 @2 (x, y, z) @2 (x, y, z) @2 (x, y, z) + + 2m @x2 @y2 @z2 ✓ ◆ +U(x, y, z) (x, y, z)=E (x, y, z) ~ 2 + U(x, y, z) (x, y, z)=E (x, y, z) 2mr  • This is a direct extension of the one-dimensional Schrödinger equation ~2 @2 (x) + U(x) (x)=E (x) 2m @x2
8. Particle in a Three-Dimensional Box • This says that the total energy is contributed by three terms on the left, each depending separately on x, y and z. Let us write E = Ex + Ey + Ez. Then this equation can be separated into three equations: 2 2 ~2 @2X(x) ~ @ Y (y) ~2 @2Z(z) = E X(x) = EyY (y) = E Z(z) 2m @x2 x 2m @y2 2m @z2 z • These obviously have the same solutions separately as our original particle in an infinite square well, and corresponding energies 2 2 2 nx⇡x nx⇡ h Xnx = Cxsin E = (nx = 1,2,3, ) L x 2mL2 ⇣ ⌘ 2 2 2 ny⇡y ny⇡ h Xny = Cysin E = (ny = 1,2,3, ) L y 2mL2 ⇣ ⌘ 2 2 2 nz⇡z nz⇡ h Xnz = Czsin E = (nz = 1,2,3, ) L z 2mL2 ⇣ ⌘
9. Energy Degeneracy • For a particle in a three-dimensional box, the allowed energy levels are surprisingly complex. To find them, just count up the different possible states. 41.2 Particle in a Three-Dimensional Box 1371 • Here are the first six states for an equal-side box: E 41.4 Energy-level diagram for a particle 2 2 6-fold degenerate in a three-dimensional cubical box. We 3⇡ h (3, 2, 1), (3, 1, 2), (1, 3, 2), 14 E label each level with the quantum numbers E = (2, 3, 1), (1, 2, 3), (2, 1, 3) 3 1, 1, 1 1,1,1 2 of the states nX, nY, nZ with that energy. 2mL not degenerate Several of the levels are degenerate (more (2, 2, 2) E 4 1, 1, 1 1 2 3-fold degenerate 11 than one state has the same energy). The (3, 1, 1), (1, 3, 1), (1, 1, 3) E 3 1, 1, 1 lowest (ground) level, nX, nY, nZ = 2 1, 1, 1 , has energy E1,1,1 = 1 + 3-fold degenerate 2 2 2 2 2 1 2 2 2 2 (2, 2, 1), (2, 1, 2), (1, 2, 2) 3E1, 1, 1 1 + 1 p U 2mL = 3p U 2mL ; we show1 the2 energies of the other 1levels as • If the length of sides of multiples2 of E>1,1,1 . > 3-fold degenerate box are different: (2, 1, 1), (1, 2, 1), (1, 1, 2) 2E1, 1, 1 2 2 2 2 2 nx ny nz ⇡ h not degenerate (1, 1, 1) E E = 2 + 2 + 2 1, 1, 1 Lx Ly Lz ! 2m (break the degeneracy) E 5 0 Since degeneracy is a consequence of symmetry, we can remove the degener- acy by making the box asymmetric. We do this by giving the three sides of the box different lengths LX , LY , and LZ . If we repeat the steps that we followed to solve the time-independent Schrödinger equation, we find that the energy levels are given by 2 2 2 2 2 nX nY nZ p U E n 1, 2, 3, ; n 1, 2, 3, ; nX,nY,nZ = 2 + 2 + 2 X = Á Y = Á LX LY LZ 2m a b 1nZ = 1, 2, 3, Á (41.17) (energy levels, particle in a three-dimensional box with2 sides of length LX , LY , and LZ) If LX , LY , and LZ are all different, the states nX, nY, nZ = 2, 1, 1 , 1, 2, 1 , and 1, 1, 2 have different energies and hence are no longer degenerate. Note 1 2 1 2 1 2 that Eq. (41.17) reduces to Eq. (41.16) if the lengths are all the same (LX = 1 2 LY = LZ = L). Let’s summarize the key differences between the three-dimensional particle in a box and the one-dimensional case that we examined in Section 40.2: • We can write the wave function for a three-dimensional stationary state as a product of three functions, one for each spatial coordinate. Only a single function of the coordinate x is needed in one dimension. • In the three-dimensional case, three quantum numbers are needed to describe each stationary state. Only one quantum number is needed in the one-dimensional case. • Most of the energy levels for the three-dimensional case are degenerate: More than one stationary state has this energy. There is no degeneracy in the one-dimensional case. • For a stationary state of the three-dimensional case, there are surfaces on 2 which the probability distribution function ƒcƒ is zero. In the one- 2 dimensional case there are positions on the x-axis where ƒcƒ is zero. We’ll see these same features in the following section as we examine a three- dimensional situation that’s more realistic than a particle in a box: a hydrogen atom in which a negatively charged electron orbits a positively charged nucleus. Test Your Understanding of Section 41.2 Rank the following states of a particle in a cubical box of side L in order from highest to lowest energy: (i) nX, nY, nZ = 2, 3, 2 ; (ii) nX, nY, nZ = 4, 1, 1 ; (iii) nX, nY, nZ = 2, 2, 3 ; (iv) nX, nY, nZ = 1, 3, 3 . ❙ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
10. Hydrogen Atom: Quantum States • Table below summarizes the quantum states of the hydrogen atom. For each value of the quantum number n, there are n possible values of the quantum number l. For each value of l, there are (2l + 1) values of the quantum number ml. • Question: How many distinct states of the hydrogen atom (n, l, ml) are there for n = 3 state? What are their energies?
11. Spectral Lines of the Hydrogen Atom E 13.6 E = 1 = [eV] n n2 n2 13.6 13.6 " = E E = m n m2 n2 ✓ ◆ ✓ ◆ Lyman series: transitions from m = 2,3, ∞ to n = 1: Ultraviolet series Balmer series: transitions from m = 3,4, ∞ to n = 2: Visible series Paschen: transitions from m = 4,5 ∞ to n = 3: Infrared series Brackett: transitions from m = 5,6 ∞ to n = 4: Infrared series 2 Pfund: transitions from m = 6,7 ∞ to n = 5: Infrared series 3
12. Spectral Lines of the Hydrogen Atom
13. Hydrogen Atom: Probability Distribution • States of the hydrogen atom with l = 0 (zero orbital angular momentum) have spherically symmetric wave functions that depend on r but not on θ or �. These are called s states. Figures below show the electron probability distributions for three of these states.
14. Quantization of Angular Momentum • In addition to quantized energy (specified by principle quantum number n), the solutions subject to physical boundary conditions also have quantized orbital angular momentum L. The magnitude of the vector L is required to obey: L = ~ l(l + 1) (l = 0, 1, 2, n-1) where l is thep orbital quantum number. • Recall that the Bohr model of the Hydrogen atom also had quantized angels moment L = nℏ, but the lowest energy state n = 1 would have L = ℏ. In contrast, the Schrodinger equation shows that the lowest state has L = 0. The wave function of this energy state is a perfectly symmetric sphere. For higher energy states, the vector L has only certain allowed directions, such that the z-component is quantized as Lz = ml~ (ml = 0, ±1, ±2, ±l)
15. Magnetic Moments and Zeeman Effect • Electron states with nonzero orbital angular momentum (l = 1,2,3, ) have a magnetic dipole moment due to electron motion. Hence these states are affected if the atom is placed in a magnetic field. The result, called Zeeman effect, is a shift in the energy of states with nonzero ml. • The potential energy associated with a magnetic dipole moment ml in a magnetic field of strength B is U = mlµBB, and the magnetic dipole moment due to the orbital angular momentum of the electron is in units of the Bohr magneton eh µB = U = mlµBB 2me
16. Electron Spin and Stern-Gerlach Experiment • The experiment of Stern and Gerlach demonstrated the existence of electron spin. The z-component of the spin angular momentum has only two possible values, corresponding to ms = +1/2 and ms = -1/2
17. Anomalous Zeeman Effect and Electron Spin • For certain atoms the Zeeman effect does not follow the simple pattern that we have described in Figure below. This is because an electron also has an intrinsic angular momentum, called spin angular momentum. 1 S = ~ s(s + 1) Sz = mz~ (mz = ) ±2 p − UmB= 2.00232sBµ (e spin magnetic interaction energy)
18. Pauli Exclusive Principle • Pauli exclusive principle states that no two electrons can occupy the same quantum-mechanical state in a given system. That is, not two electrons in an atom can have the same values of all four quantum numbers: n, l, ml, ms • For a given principal quantum number, n, there are 2n2 quantum states n l m ms Maximum Maximum number of number of electrons in electrons in the subshell the shell n=1 l=0 m=0 ms=±1/2 2 2 n=2 l=0 m=0 ms=±1/2 2 8 l=1 m=0, ±1 ms=±1/2 6 n=3 l=0 m=0 ms=±1/2 2 l=1 m=0, ±1 ms=±1/2 6 18 l=2 m=0, ±1, ±2 ms=±1/2 10
19. Periodic Table