1 CH107 Special Topics Part A: The Bohr Model of the Hydrogen Atom (First steps in Quantization of the Atom) Part B: Waves and Wave Equations (the Electron as a Wave Form) Part C: Particle (such as an Electron) in a Box (Square Well) and Similar Situations

2 A. The Bohr Model of the Hydrogen Atom

3 Setting a Goal for Part A You will learn how Bohr imposed Planck’s hypothesis on a classical description of the Rutherford model of the hydrogen atom and used this model to explain the emission and absorption spectra of hydrogen (Z = 1) and other single electron atomic species (Z = 2, 3, etc).

4 Objective for Part A Describe how Bohr applied the quantum hypothesis of Planck and classical physics to build a model for the hydrogen atom (and other single electron species) and how this can be used to explain and predict atomic line spectra.

5 Setting the Scene At the turn of the 19 th /20 th centuries, classical physics (Newtonian mechanics and Maxwellian wave theory) were unable explain a number of observations relating to atomic phenomena: Line spectra of atoms (absorption or emission) Black body radiation The photoelectric effect The stability of the Rutherford (nuclear) atom

6 The Beginnings of Bohr’s Model In 1913, Niels Bohr, a student of Ernest Rutherford, put forward the idea of superimposing the quantum principal of Max Planck (1901) on the nuclear model of the H atom. He believed that exchange of energy in quanta (E = h ) could explain the lines in the absorption and emission spectra of H and other elements.

7 Basic Assumptions (Postulates) of Bohr’s Original Model of the H Atom 1.The electron moves in a circular path around the nucleus. 2.The energy of the electron can assume only certain quantized values. 3.Only orbits of angular momentum equal to integral multiples of h/2  are allowed (m e ru = nħ, which is equation (1); n = 1, 2, 3, …) 4. The atom can absorb or emit electromagnetic radiation (h ) only when the electron transfers between stable orbits.

8 First Steps: Balance of Forces The centripetal force of circular motion balances the electrostatic force of attraction:

9 Determination of Potential Energy (PE or V) and Kinetic Energy (KE)

10 Determination of Total Energy and Calculation of u

11 Determination of Total Energy

12 Determination of Total Energy - Continued

13 Energy Level Diagram for the Bohr H Atom

14 Determination of Orbit Radius

15 Determination of Orbit Radius - Continued

16 Electronic Radius Diagram of Bohr H Atom Each electronic radius corresponds to an energy level with the same quantum number

17 The Line Spectra of Hydrogen The major triumph of the Bohr model of the H atom was its ability to explain and predict the wavelength of the lines in the absorption and emission spectra of H, for the first time. Bohr postulated that the H spectra are obtained from transitions of the electron between stable energy levels, by absorbing or emitting a quantum of radiation.

18 Electronic Transitions and Spectra in the Bohr H atom

19 Qualitative Explanation of Emission Spectra of H in Different Regions of the Electromagnetic Spectrum

20 Quantitative Explanation of Line Spectra of Hydrogen and Hydrogen-Like Species

21 …….Continued

22 Conclusion For the H atom (Z = 1), the predicted emission spectrum associated with n f = 1 corresponds to the Lyman series of lines in the ultraviolet region. That associated with n f = 2 corresponds to the Balmer series in the visible region, and so on. Likewise, the lines in all the absorption spectra can be predicted by Bohr’s equations.

23 Aftermath – the Successes and Failures of Bohr’s Model For the first time, Bohr was able to give a theoretical explanation of the stability of the Rutherford H atom, and of the line spectra of hydrogen and other single electron species (e.g. He +, Li 2+, etc). However, Bohr’s theory failed totally with two-and many-electron atoms, even after several drastic modifications. Also, the imposition of quantization on an otherwise classical description was uneasy. Clearly a new theory was needed!

24 B. Waves and Wave Equations

25 Setting a Goal for Part B You will learn how to express equations for wave motions in both sine/cosine terms and second order derivative terms. You will learn how de Broglie’s matter wave hypothesis can be incorporated into a wave equation to give Schrödinger-type equations. You will learn qualitatively how the Schrödinger equation can be solved for the H atom and what the solutions mean.

26 Objective for Part B Describe wave forms in general and matter (or particle) waves in particular, and how the Schrödinger equation for a 1- dimensional particle can be constructed. Describe how the Schrödinger equation can be applied to the H atom, and the meaning of the sensible solutions to this equation.

27

28 Basic Wave Equations

29 Characteristics of a Travelling Wave

30 Characteristics of a Standing Wave

31 Wave Form Related to Vibration and Circular Motion

32 The Dual Nature of Matter – de Broglie’s Matter Waves We have seen that Bohr’s model of the H atom could not be used on multi-electron atoms. Also, the theory was an uncomfortable mixture of classical and modern ideas. These (and other) problems forced scientists to look for alternative theories. The most important of the new theories was that of Louis de Broglie, who suggested all matter had wave-like character.

33 de Broglie’s Matter Waves De Broglie suggested that the wave-like character of matter could be expressed by the equation (5), for any object of mass m, moving with velocity v. Since kinetic energy (E k = 1/2mv 2 ) can be written as De Broglie’s matter wave expression can thus be written

34 de Broglie’s Matter Waves, Continued Since h is very small, the de Broglie wavelength will be too small to measure for high mass, fast objects, but not for very light objects. Thus the wave character is significant only for atomic particles such as electrons, neutrons and protons. De Broglie’s equation (5) can be derived from (1)equations representing the energy of photons (from Einstein – E = mc 2 – and Planck – E = hc/ ) and also (2) equations representing the electron in the Bohr H atom as a standing wave (m e vr = nh/2  ; n = 2  r)

35 The Electron in an Atom as a Standing Wave An important suggestion of de Broglie was that the electron in the Bohr H atom could be considered as a circular standing wave

36 Differential Form of Wave Equations Consider a one-dimensional standing wave. If we suppose that the value y(x) of the wave form at any point x to be the wave function  x , then we have, according to equation (4) Of particular interest is the curvature of the wave function; the way that the gradient of the gradient of the plot of  versus x varies. This is the second derivative of  with respect to x.

37 Differential Form of Wave Equations, Continued Thus Equation (9) is a second order differential equation whose solutions are of the form given by equation (7).

38 Differential Wave Equation for a One- Dimensional de Broglie Particle Wave We now consider the differential wave equation for a one-dimensional particle with both kinetic energy (E k ) and potential energy (V(x)).

39 The Schrödinger Equation for a Particle Moving in One Dimension Equation (10) shows the relationship between the second derivative of a wave function and the kinetic energy of the particle it represents. If external forces are present (e.g. due to the presence of fixed charges, as in an atom), then a potential energy term V(x) must be added. Since E(total) = E k + V(x), substituting for E k in Equation (10) gives This is the Schrödinger equation for a particle moving in one dimension.

40 The Schrödinger Equation for the Hydrogen Atom Erwin Schrödinger (1926) was the first to act upon de Broglie’s idea of the electron in a hydrogen atom behaving as a standing wave. The resulting equation (12) is analogous to equation (11); It represents the wave form in three dimensions and is thus a second-order partial differential equation.

41 The Schrödinger Equation, Continued The general solution of equations like equation (12) had been determined in the 19 th century (by Laguerre and Legendre). The equations are more easily solved if expressed in terms of spherical polar coordinates (r,  ), rather than in cartesian coordinates (x,y,z), in which case,

42 Erwin Schrödinger Students: I hope you are staying awake while the professor talks about my work! Love, Erwin

43 Spherical Polar Coordinates

44 Solutions of the Schrödinger Equation for the Hydrogen Atom The number of solutions to the Schrödinger equation is infinite. By assuming certain properties of  (the wave function) - boundary conditions relevant to the physical nature of the H atom - only solutions meaningful to the H atom are selected. These sensible solutions for  (originally called specific quantum states, now orbitals) can be expressed as the product of a radial function [R(r)] and an angular function [Y( ,  )], both of which include integers, known as quantum numbers; n, l and m (or m l ).

45 Solutions of the Schrödinger Equation for the Hydrogen Atom, Continued  (r, ,  ) = R n l (r)Y l m ( ,  ) (13) The radial function R is a polynomial in r of degree n – 1 (highest power r (n-1), called a Laguerre polynomial) multiplied by an exponential function of the type e (-r/na0) or e (-  /n), where a 0 is the Bohr radius. The angular function Y consists of products of polynomials in sin  and cos  (called Legendre polynomials) multiplied by a complex exponential function of the type e (im  ).

46 Solutions of the Schrödinger Equation for the Hydrogen Atom, Continued The principal quantum number is n (like the Bohr quantum number = 1, 2, 3,…), whereas the other two quantum numbers both depend on n. l = 0 to n - 1 (in integral values). m = - l through 0 to + l (again in integral values). The energies of the specific quantum states (or orbitals) depend only on n for the H atom (but not for many-electron atoms) and are numerically the same as those for the Bohr H atom.

47 Orbitals Orbitals where l = 0 are called ‘s orbitals’; those with l = 1 are known as ‘p orbitals’; and those with l = 2 are known as ‘d orbitals’. When n = 1, l = m = 0 only; there is only one 1s orbital. When n = 2, l can be 0 again (one 2s orbital), but l can also be 1, in which case m = -1, 0 or +1 (corresponding to three p orbitals). When n = 3, l can be 0 (one 3s orbital) and 1 (three 3p orbitals) again, but can also be 2, whence m can be –2, -1, 0, +1 or +2 (corresponding to five d orbitals).

48 Energy Levels of the H atom

49 The 1s Wave Function of H and Corresponding Pictorial Representation

50 The 2p z Wave Function of H and Corresponding Pictorial Representation

51 C. Particle in a One-Dimensional Box

52 Setting a Goal for Part C You will learn how the Schrödinger equation can be applied to one of the simplest problems; a particle in a one- dimensional box or energy well. You will learn how to calculate the energies of various quantum states associated with this system. You will learn how extend these ideas to three dimensions.

53 Objective for Part C Describe how the Schrödinger equation can be applied to a particle in a one- dimensional box (and similar situations) and how the energies of specific quantum states can be calculated.

54 Particle in a One-Dimensional Box The simplest model to which the Schrödinger equation can be applied is the particle (such as a ‘1-D electron’) in a one-dimensional box or potential energy well. The potential energy of the particle is 0 when it is in the box and  beyond the boundaries of the box; clearly the particle is totally confined to the box. All its energy will thus be kinetic energy.

55 Defining the Problem

56 Setting up the Schrödinger Equation

57 Solution of the Schrödinger Equation and use of the First Boundary Condition

58 Evaluation of the Constant k and Use of the 2 nd Boundary Condition

59 Determination of the Energy Levels

60 Determination of the Constant A

61 A Particle in a Three-Dimensional Box The arguments in the previous slides can be extended to a particle confined in a 3D box of lengths L x, L y and L z. Within the box, V(x,y,z) = 0; outside the cube it is  A quantum number is needed for each dimension and the Schrödinger equation includes derivatives with respect to each coordinate. The allowed energies for the particle are given by

62 Calculation of Energies of a Particle in a 3-D Box

63 ….Continued

64 Calculation of Energy Spacing in Different Situations

65 …Continued