Physical Chemistry 2nd Edition Chapter 13 The Schrödinger Equation Physical Chemistry 2nd Edition Thomas Engel, Philip Reid

Objectives Key concepts of operators, eigenfunctions, wave functions, and eigenvalues.

Outline What Determines If a System Needs to Be Described Using Quantum Mechanics? Classical Waves and the Nondispersive Wave Equation Waves Are Conveniently Represented as Complex Functions Quantum Mechanical Waves and the Schrödinger Equation

Outline Solving the Schrödinger Equation: Operators, Observables, Eigenfunctions, and Eigenvalues The Eigenfunctions of a Quantum Mechanical Operator Are Orthogonal The Eigenfunctions of a Quantum Mechanical Operator Form a Complete Set Summing Up the New Concepts

13.1 What Determines If a System Needs to Be Described Using Quantum Mechanics? Particles and waves in quantum mechanics are not separate and distinct entities. Waves can show particle-like properties and particles can also show wave-like properties.

Thus, Boltzmann distribution is used. 13.1 What Determines If a System Needs to Be Described Using Quantum Mechanics? In a quantum mechanical system, only certain values of the energy are allowed, and such system has a discrete energy spectrum. Thus, Boltzmann distribution is used. where n = number of atoms ε = energy

Example 13.1 Consider a system of 1000 particles that can only have two energies, , with . The difference in the energy between these two values is . Assume that g1=g2=1. a. Graph the number of particles, n1 and n2, in states as a function of . Explain your result. b. At what value of do 750 of the particles have the energy ?

Solution We can write down the following two equations: Solve these two equations for n2 and n1 to obtain

Solution Part (b) is solved graphically. The parameter n1 is shown as a function of on an expanded scale on the right side of the preceding graphs, which shows that n1=750 for .

13.2 Classical Waves and the Nondispersive Wave Equation 13.1 Transverse, Longitudinal, and Surface Waves A wave can be represented pictorially by a succession of wave fronts, where the amplitude has a maximum or minimum value.

13.2 Classical Waves and the Nondispersive Wave Equation The wave amplitude ψ is: It is convenient to combine constants and variables to write the wave amplitude as where k = 2πλ (wave vector) ω = 2πv (angular frequency)

13.2 Classical Waves and the Nondispersive Wave Equation 13.2 Interference of Two Traveling Waves For wave propagation in a medium where frequencies have the same velocity (a nondispersive medium), we can write where v = velocity at which the wave propagates

Example 13.2 The nondispersive wave equation in one dimension is given by Show that the traveling wave is a solution of the nondispersive wave equation. How is the velocity of the wave related to k and w?

Solution The nondispersive wave equation in one dimension is given by Show that the traveling wave is a solution of the nondispersive wave equation. How is the velocity of the wave related to k and w?

Example 13.2 We have

13.3 Waves Are Conveniently Represented as Complex Functions It is easier to work with the whole complex function knowing as we can extract the real part of wave function.

Standing (stationary) wave

Example 13.3 a. Express the complex number 4+4i in the form . b. Express the complex number in the form a+ib.

Solution a. The magnitude of 4+4i is The phase is given by Therefore, 4+4i can be written

Solution b. Using the relation can be written

13.2 Some Common Waves (Traveling waves) Spherical wave Planar wave Cylindrical wave

13.4 Quantum Mechanical Waves and the Schrödinger Equation The time-independent Schrödinger equation in one dimension is It used to study the stationary states of quantum mechanical systems.

13.4 Quantum Mechanical Waves and the Schrödinger Equation An analogous quantum mechanical form of time-dependent classical nondispersive wave equation is the time-dependent Schrödinger equation, given as where V(x,t) = potential energy function This equation relates the temporal and spatial derivatives of ψ(x,t) and applied in systems where energy changes with time.

13.4 Quantum Mechanical Waves and the Schrödinger Equation For stationary states of a quantum mechanical system, we have Since , we can show that that wave functions whose energy is independent of time have the form of

where {} = total energy operator or It can be simplified as 13.5 Solving the Schrödinger Equation: Operators, Observables, Eigenfunctions, and Eigenvalues We waould need to use operators, observables, eigenfunctions, and eigenvalues for quantum mechanical wave equation. The time-independent Schrödinger equation is an eigenvalue equation for the total energy, E where {} = total energy operator or It can be simplified as

Eigenequation, eigenfunction, eigenvalue The effect of an operator acting on its eigenfunction is the same as a number multiplied with that eigenfunction. There may be an infinite number of eiegenfunctions and eigenvalues.

Example 13.5 Consider the operators . Is the function an eigenfunction of these operators? If so, what are the eigenvalues? Note that A, B, and k are real numbers.

Solution To test if a function is an eigenfunction of an operator, we carry out the operation and see if the result is the same function multiplied by a constant: In this case, the result is not multiplied by a constant, so is not an eigenfunction of the operator d/dx unless either A or B is zero.

Solution This equation shows that is an eigenfunction of the operator with the eigenvalue k2.

Matrix as Quantum Mechanical Operator

Eigenvalues and eigenvector of the Matrix Operator

Exercise 不記得矩陣對角線化的同學請閱讀下述投影片或看線性代數的書： http://140.117.34.2/faculty/phy/sw_ding/teaching/chem-math1/cm07.ppt

Review of Orthogonal decomposition of vectors and functions:

Orthogonality is a concept of vector space. 13.6 The Eigenfunctions of a Quantum Mechanical Operator Are Orthogonal Orthogonality is a concept of vector space. 3-D Cartesian coordinate space is defined by In function space, the analogous expression that defines orthogonality is Orthogonormality:

Example 13.6 Show graphically that sin x and cos 3x are orthogonal functions. Also show graphically that

Solution The functions are shown in the following graphs. The vertical axes have been offset to avoid overlap and the horizontal line indicates the zero for each plot. Because the functions are periodic, we can draw conclusions about their behaviour in an infinite interval by considering their behaviour in any interval that is an integral multiple of the period.

Solution

Solution The integral of these functions equals the sum of the areas between the curves and the zero line. Areas above and below the line contribute with positive and negative signs, respectively, and indicate that and . By similar means, we could show that any two functions of the type sin mx and sin nx or cos mx and cos nx are orthogonal unless n=m. Are the functions cos mx and sin mx(m=n) orthogonal?

13.6 The Eigenfunctions of a Quantum Mechanical Operator Are Orthogonal 3-D system importance to us is the atom. Atomic wave functions are best described by spherical coordinates.

Volume element in spherical coordinates is , thus Example 13.8 Normalize the function over the interval Solution: Volume element in spherical coordinates is , thus

Solution Using the standard integral , we obtain The normalized wave function is Note that the integration of any function involving r, even if it does not explicitly involve , requires integration over all three variables.

13.4 Expanding Functions in Fourier Series 13.7 The Eigenfunctions of a Quantum Mechanical Operator Form a Complete Set The eigenfunctions of a quantum mechanical operator form a complete set. This means that any well-behaved wave function, f (x) can be expanded in the eigenfunctions of any of the quantum mechanical operators. 13.4 Expanding Functions in Fourier Series

13.7 The Eigenfunctions of a Quantum Mechanical Operator Form a Complete Set Fourier series graphs

Fourier series

The function f(x) (blue line) is approximated by the summation of sine functions (red line):

Fourier Transforms (FT)

FT in exponential form

For students who are not familiar with orthogonal expansion of functions, you may find the following ppt tutorial helpful: http://140.117.34.2/faculty/phy/sw_ding/teaching/chem-math1/cm07.ppt You should also read a chemistry math book and do some exercises for better understanding.