Ch 2. The Schrödinger Equation (S.E) - Due to the contribution of wave-particle duality, an appropriate wave equation need to be solved for the microscopic world. Erwin Schrödinger was the first to formulate such an equation - We need to be familiar with operators, eigenfunction, wavefunction, eigenvalues that are used in S.E. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 2.1 What determines if a system needs to be described using Q.M? When do we use a particle description(classical) of an atomic or molecular system and when do we use a wave (quantum mechanical) description? two criteria are used! 1) The magnitude of the wavelength of the particle relative to the dimension of the problem, a hydrogen molecule 100 pm vs. a baseball 1x 10-34 m no way to show wave character of baseball 2) The degree to which the allowed energy values form a continu ous energy spectrum Boltzmann distribution : MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry There are two limits 1) large T or small εi – εj : small energy gap(almost continuous energy), classical behavior 2) (εi – εj)/kT >> 1 : large energy gap, quantum behavior Also we can derive the internal energy by Boltzmann distribution k : the Boltzmann constant MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Relative population in the different energy levels a) Sharp energy levels b) ∆E ≈ kT : classical behavior(nearby the continuous energy) c) ∆E >> kT : quantum behavior(discrete energy) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 2.2 Classical waves and the nondispersive wave equation Example of waves a) Plane waves b) spherical waves c) cylindrical waves Wave front : surface over the maximum or minimum amplitude Direction of propagation of waves : blue arrows, perpendicular to the surface MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Mathematically, we can describe the wave by wave function. Amplitude of wave : related to position and time Position : propagation depends on wavelength λ Time : propagation depends on period T Arbitrary condition : Ψ(0,0)=0 Positions where amplitude is zero : x or t increase → wave moves in the positive x direction Graph of wave functions MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry We use another form, too. k : wave vector, ω : angular frequency, ω=2πν Choice of a zero in time or position : arbitrary(free) → We can rewrite the wave equation In this case, Ψ(0,0) ≠ 0 and φ : initial phase MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry If two or more waves in same region : interference 1) constructive interference : enhancement of amplitude 2) destructive interference : cancellation of amplitude Phase difference in interference 1) constructive interference : 2nπ → same sign 2) destructive interference : (2n+1)π → opposite sign Also, we can think about two waves same frequency, amplitude and opposite directions MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Use this formula Final result is simple than original equation position of nodes are same at all time! → ‘standing’ waves standing waves represent the stationary state(no change by time) MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Both of traveling waves and stationary waves, amplitude and distance is not independent : classical nondispersive wave eq. v : velocity of wave propagation This equation is a start point of the Schrödinger Equation. Ex) 2-2. show that traveling wave satisfies wave equation. Solution) Equating two results, v=ω/k MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 2.3 Waves represented by complex functions We know the traveling wave equation Where φ’ = φ - π/2 Use the Euler’s formula We write the wave equation in complex form Why uses a complex form? 1) All quantities can obtain by this form 2) easier than real form with differentiation and integration. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Important formulas 1) Complex number : a+ib(a,b:real) or equivalent form, 2) Complex conjugate of f(number of function) : f* substituting i to –i ex) (a+ib)* = a-ib, 3) Magnitude of f(number of function) : |f| MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 2.4 The Schrödinger Equation How we can obtain the Schrödinger Equation? → start with the classical wave equation and stationary wave Substitute the wave equation by stationary wave, we obtain Using the relations ω=2πν and νλ=v Until now, it is classical wave. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry ‘Introduce’ the quantum mechanics by de Broglie relation Momentum is related by total energy E and potential energy V(x) Substituting the wave equation by this relation Use ℏ=h/2π, we obtain the time-independent Schrödinger Eq. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry How one obtain the time-dependent Schrödinger Equation? Starting from the solution of classical wave equation On the other hand Then Wave equivalent of a free particle of energy E and momentum p moving on the x-direction MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Let’s try now, We know that Now replace Time-dependent Schrödinger Eq. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry One of our focus is stationary system. In this case, both of time-dependent and time-independent are satisfied. For stationary state, Substitute the equation, we obtain Finally, → same form as classical standing wave MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 2.5 Solving the Schrödinger Equation Key concept : operators, observables, eigenfunctions and eigenvalues Operators in a classical mechanics Ex) Velocity in Newton’s second law 1) Integrate the force acting on the particle over the interval 2) Multiply by the inverse of the mass 3) Add the quantity to the velocity at time t1 MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry How about a operators in Quantum Mechanics? → every measurable quantities(observables) have their operator each. (ex : energy, momentum, position) Notation : caret, Ô Differential equations : set of solutions Operator Ô has a set of eigenfunctions and eigenvalues Ψn : eigenfunctions, an : eigenvalues Ex) hydrogen atom eigenfunctions : each orbitals(1s, 2s, 2px, …) eigenvalues : each orbiral energies MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry We see the time-independent Schrödinger equation. This equation can be written by Important physical implication : measurement process in Q.M. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 2.6 Eigenfunctions of Q.M. operator are orthogonal Orthogonality in 3-dimensional vector space : x•y = x•z = y•z = 0 Similar, orthogonality in functional space is defined by If I = j, the integral has a nonzero value Functions can be normalized and form an orthonormal set. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry In 3-dimension, normalization must be 3-dimension. Closed-shell atoms are spherical symmetric and we normalized the wave functions in spherical coordinate Volume element in spherical coordinate : r2 sin θ dr dθ dφ, not a dr dθ dφ MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry 2.7 Eigenfunctions of Q.M. operator form a complete set completeness in 3-dimensional vector space : Any vector in 3-dimensional can be represented by linear combination of vector x, y, and z Similar, completeness in functional space : Wave function can be expanded in the eigenfunctions of any Q.M. operator It is same formalism as a Fourier series. We choose a periodic function in [-b,b] Fourier series is represented by MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry If f(x) even : cn=0, dn calculated by orthogonality Generally And this approximation is nearly exact. MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Accuracy of Fourier series a) Yellow line : real function b) red line : fourier series approximation, n=2,3,4 and 6 MS310 Quantum Physical Chemistry

MS310 Quantum Physical Chemistry Summary The time-dependent and time-independent Schrödinger equations play the role in solving quantum mechanical problems. Operators, eigenfunctions, wave functions and eigenvalues are key concepts to solve quantum mechanical wave equations. MS310 Quantum Physical Chemistry