# Quantum Mechanics: An Introduction

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Quantum Mechanics: An Introduction
Electromagnetic radiation exhibits wavelike properties which can be depicted in the following representation below: The wavelength,  (lower-case Greek letter lambda), is represented by number 1, or the peak-to-peak distance, and is generally in units of meters (although usually converted to nm). The number 2 represents the amplitude, or the height of the wave above/below the center line. Finally, the number 3 represents the node, or the point at which an electron occupying an orbital will NOT be found (to be discussed in more detail later). The number of cycles per second is called the frequency, denoted with the symbol  (the Greek letter nu), and expressed in units of s-1 of Hz (Hertz).

Both light and waves can be characterized by wavelength and frequency, and all electromagnetic radiation moves at the speed of light (c). Max Planck first analyzed data from the emission of light from hot, glowing solids. He observed that the color of the solids varied with temperature. Furthermore, Planck suggested that a relationship exists between energy of atoms in the solid and wavelength. This led to the following equation:  = c

 = c This equation has multiple interpretations. If the wavelength is long, there will be fewer cycles of the wave passing a point per second; thus, the frequency will be low. Conversely, for a wave to have a high frequency, the distance between the peaks of the wave must be small (short wavelength). In summary, an inverse relationship exists between frequency and wavelength of electromagnetic radiation as the speed of light is always constant.

Energy can be released (or absorbed) by atoms in discreet chunks (also known as quantum or photons) of some minimum size; that is, energy is quantized and emitted/absorbed in whole number units of h. Collectively, we say that: E = nh = nhc/ where n = the quantum number, h = Planck’s constant (6.626 x J s),  = frequency,  = wavelength, and c = speed of light (3.0 x 108 m/s). Energy (E) is generally expressed in units of Joules (J), where this actually implies J/atom.

Increasing Frequency () and Energy (E) Increasing Wavelength ()
Electromagnetic Spectrum Electromagnetic waves are produced by a combination of electrical and magnetic fields which are at right angles to one another (i.e. perpendicular). The various types of waves include: radiowaves, microwaves, infrared, visible light, ultraviolet, x-rays, cosmic rays, and gamma rays. All electromagnetic waves travel at the speed of light; they differ from one another by their corresponding wavelengths and frequencies. Increasing Frequency () and Energy (E) Radio Micro IR Visible UV X-ray Cosmic Gamma Increasing Wavelength ()

LIGHT ELECTROMAGNETIC RADIATION VISIBLE LIGHT IS ORDERED AS: R O Y G B I V red orange yellow green blue indigo violet 700nm nm low energy high energy long wavelength short wavelength Example: A radiation source has a frequency of 2.35 x 1014 hertz, what is the wavelength & energy associated with this light then identify the type of radiation. How much energy & what frequency is associated with a wavelength of nm? Identify this radiation.

RADIATION SOURCE TYPE OF INTERACTION WITH MATTER
X - RAY energy transfer results in an inner shell electron being ejected causing other electrons to cascade (emit) down to lower levels. High energy & short wavelength. UV/Vis energy is transferred to outershell electron resulting in the excitation of that electron which eventually cascades back down to lower(ground) energy state. Ir energy is transferred to molecule causing the molecule to vibrate; sometimes even lower energy is emitted back out or the energy is lost as heat. Microwave energy is transferred to molecule causing the molecule to vibrate and rotate; the higher initial energy is lost as vibrational and rotational energy and as heat. Low energy and long wavelength.

1. Yellow light exhibits a wavelength of approximately 570 nm. Determine the frequency of this light and the total energy of the photon being emitted in units of J and kJ/mol. 2. When an electron beam strikes a block of copper, x-rays with a frequency of 2.0 x 1018 Hz are emitted. How much energy is emitted at this wavelength by (A) an excited copper atom when it generates an x-ray photon; (B) 1.00 mol of excited copper atoms; and (C) 1.00 g of copper atoms? 3. A minimum energy of 495 kJ/mol is required to break the oxygen-oxygen bond in O2. What is the longest wavelength of radiation that possesses the necessary energy to break the bond? 4. Arrange the following types of photons of radiation in order of increasing speed: infrared radiation, visible light, x-rays, and microwaves. Which form of radiation possesses the longest wavelength? Which form of radiation possesses the largest amount of energy per photon?

KEelectron = ½ mv2 = h - ho
Photoelectric Effect Consider incident radiation which consists of a stream of photons of energy h. Once these particles strike the surface of a metal, the energy is absorbed by an electron. If the energy of the photon is less than the energy required to remove an electron from the metal, then an electron will NOT be ejected. Moreover, the energy required to remove an electron from the surface of a metal is called the work function () of the metal. However, if the energy of the photon is greater than the work function, then an electron is ejected with a kinetic energy equal to the difference between the energy of the incoming photon and the work function. That is, KEelectron = ½ mv2 = h - ho where m = mass, v = velocity, h = Planck’s constant, and  = frequency.

Photoelectric Effect These results, known as the photoelectric effect, can be summarized alongside with Einstein’s theory: 1. An electron can be driven out of the metal only if it receives at least a certain minimum energy, , from the photon during the collision. Therefore, the frequency of the radiation must have a certain minimum value if electrons are to be ejected. This minimum frequency depends on the work function and hence on the identity of the metal. 2. Provided a photon has enough energy, a collision results in the immediate ejection of an electron. 3. The kinetic energy of the ejected electron from the metal increases linearly with the frequency of the incident radiation according to the equation ½ mv2 = h - ho.

Lecture questions on EM Radiation & the Photoelectric Effect.
1. Yellow light is given off by a sodium vapor lamp used for public lighting. If the light has a wavelength of 589 nm, what is the frequency of this radiation? What is its energy? 2. What type of radiation is involved if the wavelength is 10 m? What type of radiation is involved if the frequency is 5 x 1016 Hz? What wavelength is this? 3. Calculate the ionization energy of an Iridium atom, given that radiation of wavelength 76.2 nm produces electrons with a speed of 2980 km/s.

Workshop on Photoelectric Effect
1. Calculate the ionization energy of a rubidium atom, given that radiation of wavelength 58.4 nm produces electrons with a speed of 2450 km/s. 2. The energy required for the ionization of a certain atom is 3.44 x J. The absorption of a photon of unknown wavelength ionizes the atom and ejects an electron with velocity 1.03 x 108 m/s. Calculate the wavelength of the incident of radiation. 3. The ionization energy of gold is kJ/mol. Is light with a wavelength of 225 nm capable of ionizing a gold atom in the gas phase?

de Broglie Relation and the Wave-Particle Duality of Matter
If electromagnetic radiation has a dual character, could it be that matter, which has been regarded as consisting of particles, also have wavelike properties? On the basis of theoretical observations, Louis de Broglie suggested that all particles should be regarded as having wavelike properties. Therefore, it was proposed that:  = h/p = h/mv where  = wavelength, h = Planck’s constant, p = linear momentum (in units of kg m/s), m = mass (kg), and v = velocity (m/s). BE CAREFUL: v is velocity and NOT frequency (). Beginning chemistry students often confuse these symbols! Example: If an electron travels at a velocity of x 107 m/s and has a mass of x g, what is its wavelength?

Heisenberg Uncertainty Principle
Heisenberg determined that there is a fundamental limitation to just how precisely one can know both the position and the momentum of a particle at a given time. That is, x  p > h/4 where x = the location of particle, p = linear momentum, and h = Planck’s constant. This relationship means that the more precisely we know a particle’s motion, the less precisely we can know its momentum, and vice versa. The uncertainty principle implies that we cannot know the exact path of the electron as it moves around the nucleus. It is therefore not appropriate to assume that the electron is moving around the nucleus in a well-defined orbit. Example: An electron with a mass of x kg is known to have an uncertainty of 1 pm in its position. Determine the uncertainty in its speed.

A MODEL for THE HYDROGEN ATOM Ephoton = Efinal - Einitial = hn
NIELS BOHR ( ) A MODEL for THE HYDROGEN ATOM Bohr built upon Planck’s & Einstein’s ideas about quantized energy. States: 1. The hydrogen atom has only specific allowable energy levels (quantized). 2. The atom does not radiate energy while within an energy level. 3. Electrons transition to different energy levels only by absorbing or emitting a photon whose energy equals the difference in energy between the levels. Ephoton = Efinal - Einitial = hn Limitations: 1. Failed to predict the spectrum for other atoms. (does not take into account additional nucleus-electron attractions and electron- electron repulsion) 2. Electrons do not move in “orbits”

R = Rydberg constant 1.096776 x 107 m-1
LINE SPECTRUM Bohr’s model was an attempt to explain how light was emitted when an element was vaporized and then thermally or electrically excited (the flame test or neon sign). Through experimentation it was discovered that each element has a unique line spectrum with specific wavelengths that can be used to identify it. RYDBERG EQUATION: An empirical equation used to predict the position and wavelength of the lines in a given series in a specific region of the EM spectrum. 1 = R ( ) l n2 n2 R = Rydberg constant x 107 m-1

BOHR’S MODEL & RYDBERG Combined 1913
RH = x J Rydberg constant The electron in the atom occupies specific energy levels (hn, 2hn, 3hn, etc.) This is called QUANTIZATION E = RH n2 The electron may undergo a transition from one energy level to another. Remember that the energy of emitted photon=hv=Ei - Ef hv = RH ( ) ni2 nf2 1 = RH ( ) l hc ni2 nf2 These equations can be used to determine the l or v of the hydrogen atom.

The problem with classical physics of the time was that an electron orbiting a nucleus would lose energy & eventually collapse into the nucleus. In Bohr’s model, an electron can travel around a nucleus without radiating energy. Furthermore, an electron in a given orbit has a certain definite amount of energy. The only way an electron can lose energy is by dropping from one energy level to a lower one. When this happens, the atom emits a photon of radiation corresponding to the difference in energy levels. However, electrons in higher levels cannot drop to a lower level if that level is filled. Because no electrons can move to a lower level, none of them can lose energy. The atom is energetically stable and is said to be in its ground state. Atoms can absorb energy from an outside source (such as heat from a flame or electrical energy from a source of voltage), causing one or more of the electrons within the atom to move to higher energy levels. When electrons are moved to these higher levels, the atom is said to be in an excited state. However, the atom is energetically unstable, so it will not remain in an excited state for a long period of time. Eventually, electrons return to lower levels. As they do so, energy is given off in the form of quanta.

ENERGY STATES OF THE HYDROGEN ATOM
When an electron, in its ground state, absorbs energy from a photon (called absorption), that electron is promoted to a higher energy level (called the excited state). Emission is when an electron in an excited state loses energy and returns to a lower energy level. E = x J (Z2/n2) E : the energy of the atom (derived from classical physics) Z : the charge of the nucleus n : the energy level To find the energy difference between any two levels and predict the spectral lines for the hydrogen atom: E = hn = hc l E = Ef - Ei = x J (nf ni-2) R = x J/(hc)

If ni  nf - energy is emitted
________es the electron jumps from a higher  energy level down to a lower energy ________gs level If ni  nf - energy is absorbed ________es the electron jumps from a lower  energy level to a higher one. ________gs Example: Calculate the wavelength of light that corresponds to the transition of the electron from n = 4 to n = 2 state of the hydrogen atom. Is the light emitted or absorbed? What color is it?

E = -2.178 x 10-18 J (Z2/nfinal2 - Z2/ninitial2)
Bohr Model Revisited Bohr proposed a model that included the idea that the electron in a hydrogen atom moves around the nucleus only in certain allowed circular orbits. Furthermore, Bohr concluded the following as applied to the bright-line spectrum of hydrogen: 1. Hydrogen atoms exist in only specified energy states given by the Rydberg energy equation: E = x J (Z2/nfinal2 - Z2/ninitial2) where E = Energy (J), Z = atomic number, and n = orbital level. 2. Hydrogen atoms can absorb only certain amounts of energy, and no others. 3. When excited hydrogen atoms lose energy, they lose only certain amounts of energy, emitted as photons. 4. The different photons given off by hydrogen atoms produce the color lines seen in the bright-line spectrum of hydrogen. The greater the energy lost by the atom, the greater the energy of the photon.

Workshop on Bohr’s Model
1. Determine the energy of the line in the spectrum of hydrogen that represents the movement of an electron from a Bohr orbit with n = 6 to n = 4. 2. Determine the wavelength of light that must be absorbed by a hydrogen atom in its ground state to excite it to the n = 2 orbit. NOTE: At first, Bohr’s model appeared to be very promising. The energy levels calculated by Bohr closely agreed with the values obtained from the hydrogen emission spectrum. However, when Bohr’s model was applied to atoms other than hydrogen, it did not work at all. It was therefore concluded that Bohr’s model is fundamentally incorrect for atoms with more than one electron.

Schrödinger Equation Erwin Schrödinger proposed the theory of quantum mechanics which suggested that an electron (or any other particle) exhibiting wavelike properties should be described by a mathematical equation called a wavefunction (denoted by the Greek letter psi, ). Specifically, he developed a mathematical formalism to describe the hydrogen atom as a wave. This equation (which involves differential calculus!) gives the probability of finding an electron at some point in a three-dimensional space at any given instant but offers no information about the path the electron follows (recall Heisenberg). The region in space where there is a probability of finding an electron is known as an orbital, & in the Schrödinger equation, the wavefunction is used to calculate the probability of finding an electron in space.

H is the Hamiltonian operator E is the energy of the atom
QUANTUM MECHANICS Erwin Schrodinger ( ) Schrodinger formulated the theory of wave mechanics: a description of the behavior of the tiny particles that make up matter in terms of waves. His wave equation describes the behavior of electrons in atoms. H Y = E Y Y is the wave function/atomic orbital: a mathematical description of the motion of the electron’s matter-wave in terms of position & time. H is the Hamiltonian operator E is the energy of the atom d2 Y/dx2 + d2Y/dy2 + d2Y/dz2 + (8p2me/h2)[E-V(x,y,z)] Y(x,y,z) Y2 is the probability of an electron being within a volume.

n = The principle quantum number
Quantum Mechanics In quantum mechanics, the electrons occupy specific energy levels (as in Bohr's model) but they also exist within specific probability volumes called orbitals with specific orientations in space. The electrons within each orbital has a distinct spin. n = The principle quantum number Describes the possible energy levels and pictorially it describes the orbital size. n = 1, 2, 3…. where an orbital with the value of 2 is larger than an orbital with the value of 1. 2s 1s

Quantum Mechanics l = angular momentum quantum number
Describes the "shape" of the orbital and can have values from 0 to n - 1 for each n. l = (n-1) to 0 (2 l + 1 subshells) orbital designation : s p d f shape : ml = magnetic quantum number Related to the orientation of an orbital in space relative to the other orbitals with the same l quantum numbers. It can have values between l and - l . ms = spin quantum number An electron has either +1/2 or -1/2 spin values; sometimes referred to as spin up and spin down. Too hard to draw see text

Quantum Mechanics and the Periodic Table
1. Principle Quantum Number (n): specifies the energy level of an electron and labels the shell of an atom. Represented by the period number. 2. Angular Momentum Quantum Number (l): specifies the subshell of a given shell in an atom and determines the shape of the orbitals in the subshell. Period 1 2 3 4 5 6 7

3. Magnetic Quantum Number (ml): identifies the individual orbitals of a subshell of an atom and determines the orientation in space 4a. Spin Quantum Number (ms): distinguishes the two spin states of an electron n l Orbital Designation ml # of orbital 1 2 3 4 4b. Pauli Exclusion Principle: No two electrons can have the same four sets of identical quantum numbers. Moreover, when two electrons (and NO MORE THAN TWO!) occupy the same given orbital, their spins must be different. This is due to the spin number.

Lecture Questions on Quantum Mechanics
1. Determine the quantum numbers associated with the following energy levels: a) n=2 b) n= 3 n = 5 2. Determine the sublevel names & quantum numbers: a) n = 3, l = 2 b) n = 4, l = 2 c) n = 2, l = 2 3. What is the maximum number of electrons in an atom that can have these quantum numbers? A. n = 3 B. n = 2, l = 0, ml = 0 C. n = 5, ms = +½ D. n = 2, l = 1

Workshop on quantum numbers
1. How many subshells are there for n = 6? How many total orbitals are there in the shell with n = 6? 2. Write the subshell notation and the number of electrons that can have the following quantum numbers if all the orbitals of that subshell are filled: A. n = 3; l = 2 B. n = 5; l = 0 C. n = 7; l = 1 D. n = 4; l = 3 3. How many electrons can have the following quantum numbers in an atom: A. n = 3, l = 1 B. n = 5, l = 3, ml = -1 C. n = 2, l = 1, ml = 0 4. Which of the following sets of quantum numbers {n, l, ml, ms} are allowed and which are not? For the sets of quantum numbers that are incorrect, state what is wrong. A. {2, 2, -1, -½} B. {6, 0, 0, +½} C. {5, 4, +5, +½}