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Chapter 3 Introduction to Quantum Mechanics Why Quantum Mechanics? Classical Physics is unable to offer a satisfactory explanation of phenomena in microworld, of the structure of even the simplest atom, Hydrogen. It makes no sense on atomic phenomena: potoelectronic effect, thermal radiation, optical spectra of atoms, Compton scattering, etc. Bohr-Sommerfeld theory can partially explain hydrogen atom, but it is not satisfied in theory, does not work for atoms with two or more electrons. Solution: Quantum mechanics First introduced by Schrödinger in 1926 Wave-particle duality Matter wave Correspond to a wave function r, t Of the system, to describe particle behaviors at space r and changing with time t. dV 2 Means the probability to find the particles in volume dV = dxdydz. Wavefunction properties Schrödinger equation The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the schrödinger equation will predict the distribution of results. Schrödinger equation is a fundamental assumption in Quantum Mechanics, which can not be derived from other theories. Schrödinger equation —a harmonic oscillator example For a 1D(x) free particle Wavefunction: Ψ ( x, t ) Ψ o e i ( E t P x) Ψ i i EΨ o e EΨ t Ψ P 2 Ψ oe 2 x 2 2 i ( E t P x) 2 P 2Ψ i ( Et Px ) Ψ Ψ i 2 2m x t 2 2 2 P E Ek 2m schrödinger equation For a particle with 1D(x) potential field U(x,t) 2 P E Ek U U 2m Ψ i EΨ t Ψ i P [ U ( x, t )]Ψ t 2m 2 2Ψ P2 Ψ 2 2m x 2m 2Ψ P2 2 Ψ 2 x Ψ Ψ U ( x, t )Ψ i 2 2m x t 2 2 schrödinger equation 1D → 3D schrödinger equation: Ψ [ 2 2 2 ]Ψ U ( x, y, z, t )Ψ i 2m x y z t 2 2 2 2 2 2 2 Lapalace operator: 2 2 2 x y z 2 schrödinger equation in general case: Ψ (r , t ) 2 Ψ (r , t ) U (r , t )Ψ (r , t ) i 2m t 2 Schrödinger equation is a fundamental dynamic equation of nonrelativistic quantum mechanics, which plays the role of Newton’s law in classical mechanics. Ψ (r , t ) 2 U (r , t )Ψ (r , t ) i t 2m 2 H, Hamilatonian The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wavefunction to generate the evolution of the wavefunction in time and space. The Schrödinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated. Steady Schrödinger equation Steady wavefunction : Ψ (r , t ) Φ(r )e i Et 2 2 Ψ Ψ UΨ i 2m t i 2 Et 2 2 U (r )](r )}e [ U (r )]Ψ (r , t ) {[ 2m 2m 2 i Ψ Et i EΦ(r )e t 2 2 Φ UΦ EΦ 2m or 2m Φ 2 ( E U )Φ 0 2 The structure of hydrogen atom The hydrogen atom consists of a single proton surrounded by a single electron. It is thus the simplest of all atoms. The proton may be thought to be approximately at rest at the origin of the coordinate (the center of the hydrogen atom) because proton is about 1836 times heavier than electron. The Coulomb attractive force works between the proton and the electron. Its potential is written: e2 V (r ) 40 r 1 where r is the distance between the proton and the electron. The (nonrelativistic) Schrödinger equation describing the motion of the electron takes the form: 2 2 1 e2 E 40 r 2m The Schrödinger equation and its solution The polar coordinate (r,,) shown in the following is more convenient than the Cartesian coordinate (x,y,z). The Schrödinger equation and its solution We solve the Schrödinger equation by setting the boundary condition that the wave function should be smoothly continuous at every point of the coordinate space and should converge to 0 at the infinitely long distance. Then we have a set of discrete energy eigenvalues and the corresponding eigenstates. The details of the method to solve it is omitted here. If you want to study them, please refer to some other textbooks of quantum mechanics. The wave functions of the eigenstates is expressed as (r , , ) Rnl (r )Ylm ( , ) (r , , ) Rnl (r )Ylm ( , ) the part Rnl(r) is the radial wave function which is specified by a set of integers, n and l. n and l are quantum numbers, which characterize the eigenstates. n = 1,2,3,…; l = 0,1,2,…; l n-1 The part Ylm(,) denotes the angular wave function. It describes the revolving state of the electron around the coordinate origin (proton), which is specified by a set of quantum numbers (integers), l and m: m l, i.e., m = -l, -l +1, …, l - 1, l Quantum numbers n: principal quantum number, n = 1, 2, 3, …; l: orbital quantum number, l = 0, 1, 2, …, n-1; m: magnetic quantum number, m = -l, -1+1, …, l-1, l. The energy eigenvalues of hydrogen atom are determined only by the quantum number n The ground state: The abscissa denotes the position coordinate of the electron (the distance between the proton and electron), r , in units of the Bohr radius , where 40 2 10 a0 0 . 529 10 m 2 me The energy is quantised; En is continues when n The orbital quantum number l expresses the speed of the revolution of the electron, i.e. the magnitude of the angular momentum of the electron; and the magnetic quantum number m represents the orientation (direction) of angular momentum vector. The angular momentum of the electron is quantised: L h l (l 1) 2 l (l 1) The orientation of the angular momentum is also quantised, i.e., the component in z direction is quantised: h LZ m m 2 For example: l 2 B(z) L l (l 1) 2(2 1) 6 LZ m m 0 , 1, 2 , , l L 6 Lz 2 m=1 0 m=0 m = -1 2 LZ 0, , 2 m=2 m = -2 These result implies that not only energy but also angular momentum and its orientation are quantized in quantum mechanics. This was confirmed by the Stern-Gerlach experiment (1922). Needless to say, this also originates from the particle-wave duality of electrons. And this can never understood by the classical theory. The probability to find an electron at the position r from the center— the probability density in the space: r Rnl r 2 2 2 0 r Rnl r dr 1 2 The atomic models Plum-pudding model by Thomson Electron cloud model Planet model by Rutherford Bohr’s model Atomic orbitals n=1,l=0 n=2,l=0 n=2,l=1 n=3,l=0 n=3,l=1 n=3,l=2 m=0 m=1 m=2 Atomic orbitals n=4,l=0 n=4,l=1 n=4,l=2 n=4,l=3 m=0 m=1 m=2 m=3 The probability density of the electrons of H atom n=1 l=0 m=0 l=0 m=0 n=2 l=1 l=0 n=3 l=1 l=2 n=6 l=3 n = 11 l = 6 m=0 & m = ±1 m=0 m=0 & m = ±1 m=0 & m = ±1 & m = ±2 m=0 m = ±3 The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 1, l = 0, m = 0, spherically symmetrical distributions The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 2, l = 0, m = 0, spherically symmetrical distributions The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 2, l = 1, m = 0, Dumbbell shaped distribution along one axis The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 2, l = 1, m = ±1, Dumbbell shaped distribution along one axis The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 3, l = 0, m = 0, spherically symmetrical distributions The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 3, l = 1, m = 0, The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 3, l = 1, m = ±1, The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 3, l = 2, m = 0, The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 3, l = 2, m = ±1, The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 3, l = 2, m = ±2, The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 6, l = 3, m = 0, The colors in the plots of the probability distributions vary from blue to red corresponding to the increase of the probability from small (zero) to large values. n = 11, l = 6, m = ±3,