Introduction to Quantum Physics and Models of Hydrogen Atom

Introduction to Quantum Physics and Models of Hydrogen Atom Tien-Tsan Shieh Department of Applied Math National Chiao-Tung University November 7, 2012 Physics and Models of Hydrogen November Atom 7, 2012 1 / 54

Outline Atomic Model Schrodinger equation The hydrogen atom Angular Momentum Elementary Theory of Radiation Spin quantum number Physics and Models of Hydrogen November Atom 7, 2012 2 / 54

Atomic Model Physics and Models of Hydrogen November Atom 7, 2012 3 / 54

The idea of molecules and atoms The idea of molecular and atomic nature of matter was proposed long before the real existence of molecules and atoms could be proved. In 1869, D. I. Mendeleev conceived the periodic table of elements. Avogadro s hypothesis (1776-1856) states equal volumes of gases at fixed pressure and temperature must contain equal numbers of molecules. In early 19 century, the botanist R. Brown observed the distinctive random motion of pollen suspended in a fluid. Einstein s treatment of Brownian motion lead to determined the Avogadro s number N A = 6.60221367 10 23 molecules/mole. Physics and Models of Hydrogen November Atom 7, 2012 4 / 54

Thomson s atomic model(1903) Thomson s model predict the frequency of electromagnetic wave arising from oscillation of electrons. However it did not fit with the result from observations. Direct test of Thomson s model were carried out by means of Rutherford α-particle scattering experiment. Rutherford scattering in 1911 led to the development of the Rutherford model (planetary model) of the atom, and eventually to the Bohr model. Physics and Models of Hydrogen November Atom 7, 2012 5 / 54

Rutherford s atomic model In 1911, Rutherford proposed that the massive positive component of the atom was contained within a very small region of the atomic volume, called the atomic nucleus. A scale of size could be determined for the nucleus by α-particle scattering. The atomic radius is of order 10 10 m, and the nuclear radius was found from the experiment to be of order 10 14 m. The Rutherford atom resembled the solar system. The atomic electrons were supposed to move in stable orbit around the nucleus, however Rutherford s model fail to explain this. Physics and Models of Hydrogen November Atom 7, 2012 6 / 54

The Bohr model of One-electron atom In 1913, Bohr proposed his theory of one-electron atom: 1 Electrons still obey the classical physic law: 2 1 e 4πɛ 0 r 2 = me v 2, K = 1 r 2 me v 2 = 1 2 e 8πɛ 0 r, V = 1 2 e 4πɛ 0 r, E = K + V = 1 1 8πɛ 0 r 2 Assume the existence of stationary state of electron orbits on which electrons do not emit radiation. L = r p, m e vr = n, v = n m e r r n = 4πɛ 0 2 m e e 2 n 2 = a 0 n 2, a 0 = 0.0529 nm me e4 1 E n = 32π 2 ɛ 2 0 2 n 2 = 13.6 ( 2π c 1 ev, hf = n2 λ = En 1 En 2 = 13.6eV n2 2 1 ) n1 2 (n 1 n 2 ) 3 The atom may undergo non-classical transition of these allowed state to another by emitting or absorbing a single quantum of electromagnetic radiation. ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom 7, 2012 7 / 54

Spectrum of Hydrogen Atom The Rydberg formula: 1 λ = R( 1 n 1 2 n ) 2 R = 1.097373 10 7 m 1 where n is the finial level and n is the initial level and R is called the Rydberg constant. ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom 7, 2012 8 / 54

The Limitation of the Bohr model of one-electron atom The Bohr model gives a clear picture to the motion of electron in atom which motive the ideas of stationary orbits, discrete energy of spectrum, quantization of angular momentum. However, the Bohr s model is not a complete model which only valid for one-electron atom and is difficult to extend to two-elctrons or many-electrons atoms. For many-electron atoms, the model can not count the effect of interaction between electrons. In the point view of modern quantum mechanics, the Bohr s model does not satisfy the uncertain principle. Physics and Models of Hydrogen November Atom 7, 2012 9 / 54

Schrödinger Equation Physics and Models of Hydrogen November Atom7, 2012 10 / 54

Historical development of Quantum mechanics In 19th century, Hamilton had investigated the connection between the equations of geometrical optics and wave optics. His work inspired Schrodinger who argued that classical particle dynamics should bear a similar relation to a proposed wave theory for a quantum particle. In 1924, de Broglie proposed that matter should possess both particle and wave characteristics. ν (frequancy) = E h, λ (wavelength) = h p In 1926, Schrodinger adopted de Broglie s ideas of matter waves and found the evolution equation of these matter waves. In 1926, Heisenberg s picture of quantum mechanics gave a representation of measurable quantities by means of matrix. In 1926, M. Born gave a probabilistic interpretation of solutions to Schrodinger s wave equation. Physics and Models of Hydrogen November Atom7, 2012 11 / 54

The Schrodinger equation Conservation of energy: p 2 2m 0 + V = E Planck-Einstein and de Broglie formulas: Wave packet: The dissipative relation ψ(x, t) = E = ω, 2 k 2 + V = ω, 2m 0 The Schrodinger equation: i ( t ψ = 2 p = k A(k)e i(kx ωt) dk p i x, ) x 2 + V 2 2m 0 }{{} Hamiltonian H E i t Physics and Models of Hydrogen November Atom7, 2012 12 / 54 ψ.

The probabilistic interpretation M. Born proposed we took ψ to be a mathematical wave function instead and identified ψ 2 as the basic observable quantity of the matter wave. Probability density: P(x, t) = Ψ(x, t) 2 Conservation law of probability: Ψ(x, t) 2 dx = 1 P(x, t) + j(x, t) = 0 t where the probability current density is j(x, t) = 2im (Ψ Ψ Ψ Ψ ). Physics and Models of Hydrogen November Atom7, 2012 13 / 54

Stationary States The 1D Schrodinger equation: We seek a solution taking a form We find ψ(x) and f (t) satisfies 2 2m 0 2 x 2 Ψ + V Ψ = i t Ψ. ψ(x, t) = ψ(x)f (t). 2 2m 0 d 2 dx 2 ψ + V ψ = Eψ We find i df dt = Ef. Ψ(x, t) = ψ(x)e iet/. Physics and Models of Hydrogen November Atom7, 2012 14 / 54

The Hydrogen Atom Physics and Models of Hydrogen November Atom7, 2012 15 / 54

The Hydrogen Atom The Hamiltonian of the hydrogen atom: H = The modern hydrogen model should have following features: proton Electromagnetic Interaction electron Quantum mechanics p2 + V = 2 2 e2 2m 0 2m 0 r Schrodinger equation for the hydrogen atom: i ψ t = Hψ i ψ ) ( t = 2 2 e2 ψ 2m 0 r Physics and Models of Hydrogen November Atom7, 2012 16 / 54

The Schrodinger equation in the spherical coordinate In the spherical coordinate (r, θ, φ), the Laplacian becomes 2 ψ = 1 r 2 r 2 (rψ) + 1 r 2 sin θ and Schrodinger equation becomes i ψ [ t = 2 1 2 2m 0 r r (rψ) + 1 2 r 2 Suppose that ψ take the form ( 1 sin θ ( sin θ ψ ) + θ θ ( sin θ ψ ) + 1 θ θ sin 2 θ ψ = U(r) P(θ)Q(φ)T (t). r Plugging into the Schrodinger equation, we find [ i dt T dt = 2 1 d 2 U 2m U dr + 1 ( ( 1 d sin θ dp 2 r 2 sin θ P dθ dθ ) + 1 2 ψ r 2 sin θ 2 φ 2 1 sin 2 θ Q )] 2 ψ e2 φ 2 r ψ. )] d 2 Q e2 dφ 2 r }{{} V (r). Physics and Models of Hydrogen November Atom7, 2012 17 / 54

Eigenvalue Problems Sturm-Liouville Problems [ i dt T dt = 2 1 d 2 U 2m 0 U dr + 1 ( 1 2 r 2 sin θ P ( d sin θ dp ) + dθ dθ By using the separation of variables, we obtain 1 sin θ ( d sin θ dp ) dθ dθ Some conditions for P, Q, U: dt dt = i E T d 2 Q dφ 2 = λq λ sin 2 θ P = αp 1 sin 2 θq d 2 U dr + α 2 r U = 2m0 (E V (r)) U 2 2 Q(0) = Q(2π) P(θ) finite at θ = 0, π U(r) bounded for r [0, ] )] d 2 Q + V (r). dφ 2 Physics and Models of Hydrogen November Atom7, 2012 18 / 54

Equations for T (t) and Q(φ) The equation for T (t): dt dt = i E T T (t) = e iet The equation for Q(φ): { d 2 Q dφ 2 = λq Q(0) = Q(2π) The eigenvalues are λ = m 2 for any integer m and the corresponding eigenfunctions are Q(φ) = e ±imφ. This m is called the magnetic quantum number. Physics and Models of Hydrogen November Atom7, 2012 19 / 54

Equation for P(θ) The equation for P(θ): The eigenvalue α is 1 sin θ d dθ P(θ) α = l(l + 1) ( ) sin θ dp dθ m 2 sin 2 θ P = αp finite at θ = 0, π This l is called the orbital quantum number. The corresponding eigenfunction P(θ) is where P m l where l is an integer m P(θ) = P m l (cos θ) (s) is the associated Legendre function defined by P m l (s) = ( 1)m 2 l l! d ds ((1 s2 ) dpm l ds ) + (1 s 2 ) m/2 d l+m ds l+m [(s2 1) l ] (l(l + 1) m2 1 s 2 When m = 0, P l (s) = P 0 l (s) are called the Legendre functions. ) P m l = 0. Physics and Models of Hydrogen November Atom7, 2012 20 / 54

Legendre Polynomials (1 2µs + s 2 ) 1/2 = l=0 P l (µ)s l. 1 2 P l1 (µ)p l2 (µ) dµ = 1 2l + 2 δ l 1,l 2 1 P m 2(l + m)! k (µ)pm l (µ) dµ = 1 (2l + 1)(l m)! δ k,l 1 Pl m Pl n 0 (m n) 1 (1 µ 2 ) dµ = (l+m) (m = n 0) m(l m)! (m = n = 0) ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom7, 2012 21 / 54

Spherical Harmonic Functions(I) The spherical harmonic function are defined as [ Yl m 2l + 1 (θ, φ) = 4π where l m l, of θ and φ). (l m)! (l + m)! ] 1/2 Pl m (cos θ)e imφ 0 l <.Polar plots are shown of a few low-order real spherical harmonics (function Physics and Models of Hydrogen November Atom7, 2012 22 / 54

Spherical Harmonic Functions(II) The spherical harmonic function are defined as [ 2l + 1 Yl m (θ, φ) = 4π where l m l, 0 l <. (l m)! (l + m)! ] 1/2 Pl m (cos θ)e imφ Physics and Models of Hydrogen November Atom7, 2012 23 / 54

Spherical Harmonic Functions (III) We have the normalization and orthogonality condition: 2π 0 π0 Y l m (θ, φ)y lm (θ, φ) sin θ dθdφ = δ l l δ m m. Physics and Models of Hydrogen November Atom7, 2012 24 / 54

Spherical Harmonic Functions (IV) Nodal sets ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom7, 2012 25 / 54

Spherical Harmonic Functions (V) Physics and Models of Hydrogen November Atom7, 2012 26 / 54

The Radial equation(i) The radial equation is ( d 2 l(l + 1) + dr 2 r 2 2m 0 e 2 2 2m 0 r 2 E ) U = 0. Here, we are seeking the bound state of hydrogen. Therefore, the corresponding eigenvalues should be negative. i.e. E = E. Introducing ρ 2κr where κ defined by 2 κ 2 2m 0 = E. Also define ( ) 2 ξ 2 1 = = R κa 0 E, R = 2, a 2m 0a0 2 0 = 2 m 0e 2 where R is the Rydberg constant and a 0 is the Bohr radius. We obtain ( d 2 u l(l + 1) ξ u + dρ2 ρ 2 ρ 1 ) u = 0 4 where u(ρ) = U(r). Physics and Models of Hydrogen November Atom7, 2012 27 / 54

The Radial equation(ii) For large value of ρ, we find that d 2 ( u l(l + 1) ξ dρ2 ρ 2 u + ρ 1 ) u = 0. 4 d 2 u dρ u 2 4 = 0 u A } e ρ/2 {{} +B }{{} e ρ/2. 0 as ρ as ρ For small value of ρ, we find that d 2 u l(l + 1) u = 0 u A ρ l dρ2 ρ 2 }{{} Therefore, we seek solution of form with F (ρ) = i=0 C iρ i satisfying [ρ d 2 dρ 2 + (2l + 2 ρ) d dρ u(ρ) = e ρ/2 ρ l+1 F (ρ) as ρ 0 ] (l + 1 ξ) F (ρ) = 0. +B ρ l+1 }{{}. 0 as ρ 0 Physics and Models of Hydrogen November Atom7, 2012 28 / 54

The Radial equation(iii) We will find the recurrence relation This implies that So C i+1 = C i+1 C i i (i + l + 1) ξ (i + 1)(i + 2l + 2) Γ ilc i. F (ρ) = C i ρ i e ρ. i=0 u(ρ) e ρ/2 ρ l+1 e ρ = e ρ/2 ρ l+1 diverges for large ρ. To obtain a finite wave function, for any given l, C i must terminate at some finite value i, called i max. Γ imax l = 0 i max + l + 1 = ξ n This n is called the principle quantum number. n = 1, 2, 3,.... Physics and Models of Hydrogen November Atom7, 2012 29 / 54

The Radial equation(iv) The eigenvalues are ξ 2 R E ξ 2 n = n 2 = R E n E n = R n 2. The eigenfunctions are u nl (ρ) = e ρ/2 ρ l+1 F nl (ρ) n l 1 u nl (ρ) = A nl e ρ/2 ρ l+1 i=0 C i ρ i where C i+1 = Γ il C i, ρ 2κ n r, κ n = Z a 0 n. The polynomial F nl (ρ) is called associated Laguerre polynomial, L 2l+1 n l 1. Note that R(r) = u(2κ nr)/r. Physics and Models of Hydrogen November Atom7, 2012 30 / 54

Degeneracy of E n Degeneracy of E n = n 1 l=0 (2l + 1) = n2 n = 1, 2, 3,.... Since i max 0, we have l n 1. For a given principle quantum number n, the orbital quantum number l = 0, 1, 2, 3..., n 1 l max = n 1. }{{} s,p,d,forbital Eigenenrgy E n only depends on the principle quantum number n and n distinct orbital states are degenerate. The complete eigenstates contain Yl m (θ, φ). For each l, there are 2l + 1 values of m l. m l = l,..., l. The normalized eigenfunction of Hydrogen ψ nlm (r, θ, φ) = (2κ) 3 2 Anl ρ l e ρ 2 Fnl (ρ)y m l where A nl = (n l 1)! 2n[(n+l)] 3. (θ, φ) = R nl (r)y m l (θ, φ) Physics and Models of Hydrogen November Atom7, 2012 31 / 54

The Hydrogen atom (VIII) Physics and Models of Hydrogen November Atom7, 2012 32 / 54

The Hydrogen atom (IX) Physics and Models of Hydrogen November Atom7, 2012 33 / 54

The Hydrogen atom (X) Physics and Models of Hydrogen November Atom7, 2012 34 / 54

The Hydrogen atom (XI) Physics and Models of Hydrogen November Atom7, 2012 35 / 54

Angular Momentum Physics and Models of Hydrogen November Atom7, 2012 36 / 54

Angular Momentum (I) In classical mechanics, angular momentum of a particle is defined by where p is its linear momentum. In cartesian coordinate, L = r p L x = yp z zp y L y = zp z xp z L z = xp y yp x In quantum mechanics, the linear momentum operator is ˆp = i and the angular momentum is defined by L = i r. In cartesian coordinate, In spherical coordinate: ( L x = i y z z ) y ( L y = i z x x ) z ( L z = i x y y ) x ( L x = i sin φ ) + cot θ cos φ θ φ ( L y = i cos φ ) + cot θ sin φ θ φ L z = i φ ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom7, 2012 37 / 54

Angular Momentum (II) Because of [ˆx, ˆp x] = i,..., we have The total angular momentum is the vector operator ˆL = e x ˆLx + e y ˆLy + e z ˆLz. Define We have ˆL 2 = ˆL 2 x + ˆL 2 y + ˆL 2 z. i ˆL = ˆL ˆL. [ˆL x, ˆL 2 ] = [ˆL y, ˆL 2 ] = [ˆL z, ˆL 2 ] = 0 [ˆL, ˆL 2 ] = 0 The operator L 2 in the spherical coordinate: L 2 = 2 [ 1 sin θ θ ( sin θ θ ) + 1 sin 2 θ ] 2 φ 2 Physics and Models of Hydrogen November Atom7, 2012 38 / 54

Laplacian in the spherical coordinates The Laplacian operator: 2 = 1 r 2 r 2 r + 1 r 2 ( 1 sin θ θ sin θ θ + 1 sin 2 θ ) 2 φ 2 Radial momentum: p r 1 2 ( 1 r r p + p r 1 r ) where 1 r p = i 1 r = i and p r 1 = i e r r r r r. So that p r f (r) = i ( ) ( f 2 r + er f = i r + f ) r The free-particle Hamiltonian: H = p2 2m = 2 2 2m = p2 r 2m + L2 2mr. 2 Physics and Models of Hydrogen November Atom7, 2012 39 / 54

( L x = i sin φ ) + cot θ cos φ θ φ ( L y = i cos φ ) + cot θ sin φ θ φ L z = i φ [ L 2 = 2 1 sin θ θ ( sin θ ) + 1 θ sin 2 θ 2 ] φ 2 Since ˆL 2 and ˆL z commute, they have same eigenfunctions Yl m. Let the integer l and m are related to the eigenvalues of ˆL 2 and ˆL z : L 2 Yl m = 2 l(l + 1) Yl m (l = 0, 1, 2,... ) L z Yl m = m Yl m (m = l,..., l) ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom7, 2012 40 / 54

Elementary Theory of Radiation Physics and Models of Hydrogen November Atom7, 2012 41 / 54

Elementary Theory of Radiation (I) The eigenstates ψ nlm of the hydrogen atom are stable against radiation. r = ψ nlm r ψ nlm = r ψ nlm 2 dr = 0. If we find r e z cos ωt, then the electron is doing linear simple harmonic oscillation, which is called dipole radiation. In the Bohr theory of radiation, radiation occurs when a photon is emitted during a transition from one eigenenergy state to a lower one. Suppose ψ = aψ n + bψ n and a 2 + b 2 = 1 where a(t), b(t) are time-dependent. We will find r = aψ n + bψ n r aψ n + bψ n r(t) = a be i(en E n )t/ ϕn r ϕ n +b ae i(e n En)t/ ϕn r ϕ n r(t) = 2 a b ϕ n r ϕ n cos(ω nn t + δ) = 2 r nn cos(ω nn t + δ) where ω nn is the Bohr frequency ω nn = E n E n. The average radiated power of the dipole radiation is P = 1 3 ω4 c 3 d 2 where the dipole moment d = er 0 = 2er nn with r = r 0 cos ωt. ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom7, 2012 42 / 54

Elementary Theory of Radiation (II) Selection rules: From r nn 0, we obtain the following rules: l = l l = ±1, m l = m m = 0, ±1. Example: The transition 3S 1S ( l = 0) is forbidden, as is 3D 2S ( l = 2). For a transition l = 1, the law of conservation of angular momentum implies the electromagnetic field (i.e. the photon) carries away angular momentum. It turns out photons have angular momentum 1 and are bosons. There are no restriction on an atomic transition corresponding to change in the principle quantum ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physicsnumber and Models n. of Hydrogen November Atom7, 2012 43 / 54

The Hint of the Fourth Quantum Number Spin Physics and Models of Hydrogen November Atom7, 2012 44 / 54

Atomic Orbits Physics and Models of Hydrogen November Atom7, 2012 45 / 54

Periodic Table(I) Physics and Models of Hydrogen November Atom7, 2012 46 / 54

Periodic Table(II) Physics and Models of Hydrogen November Atom7, 2012 47 / 54

Historical background The Schrodinger theory of the atom is based on a nonrelativistic treatment of electron motion. The effect of relativity are appropriately ignored in first approximation, but the resulting picture of the atom is incomplete until such behavior is taken into account. In 1896, (normal) Zeeman effect was observed by P. Zeeman. Improvements in spectroscopic revealed the splitting of lines into multiplets of closely spaced without the influence of an applied magnetic field. This is called fine structure. Pauli was able to explain the occurrence of multiplets by letting the electron have a nonclassical two-valued property, which appeared as a fourth quantum number spin. Physics and Models of Hydrogen November Atom7, 2012 48 / 54

The Zeeman effects Zinc Singlet Normal Triplet Sodium principal doublet Anomalous patterns Zinc sharp triplet Anomalous patterns Selection rules: From r nn 0, we obtain the following rules: l = l l = ±1, m l = m m = 0, ±1. ien-tsan Shieh (Department of Applied MathNational Introduction Chiao-Tung to QuantumUniversity) Physics and Models of Hydrogen November Atom7, 2012 49 / 54

The Stern-Gerlach experiment In 1922, O. Stern and W. Gerlach examines the dynamics of a magnetic dipole in a nonuniform magnetic field. Physics and Models of Hydrogen November Atom7, 2012 50 / 54

The Hint of the Fourth Quantum Number Periodic table The Zeeman effect fine structure of spectrum The Stern-Gerlach experiment These phenomena indicate that the principle quantum number n, the orbital quantum number l, the magnetic quantum number are not complete to distinguish different states of quantum systems. Therefore, Pauli introduce the fourth quantum number spin angular momentum. Physics and Models of Hydrogen November Atom7, 2012 51 / 54

The Pauli equation The Pauli equation, also known as the Schrodinger-Pauli equation, is the formulation of the Schrodinger equation for spin- 1 particles which takes into account the interaction of 2 the particle s spin with the electromagnetic field. The Pauli equation was proposed by Pauli in 1927. It is non-relativistic version of the Schrodinger equation. The Pauli equation: i t ψ = 1 ( ) (σ (p qa)) 2 + qφ ψ } 2m {{} Hamiltonian H where σ = (σ x, σ y, σ z) is a vector of 2 2 Pauli matrices. ( ) ( ) ( 0 1 0 i 1 0 σ x =, σ 1 0 y =, σ i 0 z = 0 1 More precisely, i ( ψ1 t ψ 2 t ) = 1 2m ( 3 ( (σ n i )) ) 2 qa n x n n=1 i ( (p qa) 2 ) t ψ ± = + qφ ˆ1 ψ ± 2m }{{} Schrödinger equation ) ( ) + qφ ψ1 ψ 2 q 2m σ B ψ ± }{{} Stern-Gerlach term Physics and Models of Hydrogen November Atom7, 2012 52 / 54

Summary It take time and many people s efforts to build a right model for physics which consists with experimental observation. We learn the Hydrogen models. Some special functions, such as Legendre polynomials, spherical harmonic functions, are related to symmetry of physical systems. Atomic orbitals are eigenfunctions of quantum system. The four quantum numbers: The principle quantum number: n = 1, 2, 3,.... The angular (orbital) quantum number: The magnetic quantum number: l = 0, 1, 2, n 1 m l = l, l + 1,..., 0,..., l 1, l. The spin projection quantum number: An electron has spin s = 1 2. ms will be ± 1 2. m s = s, s + 1,..., 0,..., s 1, s. Physics and Models of Hydrogen November Atom7, 2012 53 / 54

Thank you for your attention! Physics and Models of Hydrogen November Atom7, 2012 54 / 54