1 The Failures of Classical Physics Observations of the following phenomena indicate that systems can take up energy only in discrete amounts (quantization of energy): Black-body radiation Heat capacities of solids Atomic spectra

2 Black-body Radiation Hot objects emit electromagnetic radiation An ideal emitter is called a black-body The energy distribution plotted versus the wavelength exhibits a maximum. –The peak of the energy of emission shifts to shorter wavelengths as the temperature is increased The maximum in energy for the black-body spectrum is not explained by classical physics –The energy density is predicted to be proportional to -4 according to the Rayleigh-Jeans law –The energy density should increase without bound as  0

3 Black-body Radiation – Planck’s Explanation of the Energy Distribution Planck proposed that the energy of each electromagnetic oscillator is limited to discrete values and cannot be varied arbitrarily According to Planck, the quantization of cavity modes is given by: E=nh (n = 0,1,2,……) –h is the Planck constant – is the frequency of the oscillator Based on this assumption, Planck derived an equation, the Planck distribution, which fits the experimental curve at all wavelengths Oscillators are excited only if they can acquire an energy of at least h according to Planck’s hypothesis –High frequency oscillators can not be excited – the energy is too large for the walls to supply

4 Heat Capacities of Solids Based on experimental data, Dulong and Petit proposed that molar heat capacities of mono-atomic solids are 25 J/K mol This value agrees with the molar constant-volume heat capacity value predicted from classical physics ( c v,m = 3R) Heat capacities of all metals are lower than 3R at low temperatures –The values approach 0 as T  0 By using the same quantization assumption as Planck, Einstein derived an equation that follows the trends seen in the experiments Einstein’s formula was later modified by Debye –Debye’s formula closely describes actual heat capacities

5 Atomic Spectra Atomic spectra consists of series of narrow lines This observation can be understood if the energy of the atoms is confined to discrete values Energy can be emitted or absorbed only in discrete amounts A line of a certain frequency (and wavelength) appears for each transition

6 Wave-Particle Duality Particle-like behavior of waves is shown by –Quantization of energy (energy packets called photons) –The photoelectric effect Wave-like behavior of waves is shown by electron diffraction

7 The Photoelectric Effect Electrons are ejected from a metal surface by absorption of a photon Electron ejection depends on frequency not on intensity The threshold frequency corresponds to h o =  –  is the work function (essentially equal to the ionization potential of the metal) The kinetic energy of the ejected particle is given by: ½mv 2 = h -  The photoelectric effect shows that the incident radiation is composed of photons that have energy proportional to the frequency of the radiation

8 Diffraction of electrons Electrons can be diffracted by a crystal –A nickel crystal was used in the Davisson-Germer experiment The diffraction experiment shows that electrons have wave-like properties as well as particle properties We can assign a wavelength,, to the electron = h/p (the de Broglie relation) A particle with a high linear momentum has a short wavelength Macroscopic bodies have such high momenta (even et low speed) that their wavelengths are undetectably small

Chapter 119 The Schrödinger Equation Schrödinger proposed an equation for finding the wavefunction of any system The time-independent Schrödinger equation for a particle of mass m moving in one dimension (along the x-axis): (-h 2 /2m) d 2  /dx 2 + V(x)  = E  –V(x) is the potential energy of the particle at the point x –h = h/2  –E is the the energy of the particle

Chapter 1110 The Schrödinger Equation The Schrödinger equation for a particle moving in three dimensions can be written: (-h 2 /2m)  2  + V  = E  –  2 =  2 /  x 2 +  2 /  y 2 +  2 /  z 2 The Schrödinger equation is often written: H  = E  –H is the hamiltonian operator –H = -h 2 /2m  2 + V

11 The Born Interpretation of the Wavefunction Max Born suggested that the square of the wavefunction,  2, at a given point is proportional to the probability of finding the particle at that point –  *  is used rather than  2 if  is complex –  * =  conjugate In one dimension, if the wavefunction of a particle is  at some point x, the probability of finding the particle between x and (x + dx) is proportional to  2 dx –  2 is the probability density –  is called the probability amplitude

Chapter 1112 The Born Interpretation, Continued For a particle free to move in three dimensions, if the wavefunction of the particle has the value  at some point r, the probability of finding the particle in a volume element, d , is proportional to  2 d  –d  = dx dy dz –d  is an infinitesimal volume element P   2 d  –P is the probability

Chapter 1113 Normalization of Wavefunction If  is a solution to the Schrödinger equation, so is N  –N is a constant –  appears in each term in the equation We can find a normalization constant, so that the probability of finding the particle becomes an equality P  (N  *)(N  )dx –For a particle moving in one dimension  (N  *)(N  )dx = 1 –Integrated from x =-  to x=+  –The probability of finding the particle somewhere = 1 –By evaluating the integral, we can find the value of N (we can normalize the wavefunction)

14 Normalized Wavefunctions A wavefunction for a particle moving in one dimension is normalized if   *  dx = 1 –Integrated over entire x-axis A wavefunction for a particle moving in three dimensions is normalized if   *  d  = 1 –Integrated over all space

15 Spherical Polar Coordinates For systems with spherical symmetry, we often use spherical polar coordinates ( r, , and  ) –x = r sin  cos  –y = r sin  sin  –z = r cos  The volume element, d  = r 2 sin  dr d  d  To cover all space –The radius r ranges from 0 to  –The colatitude, , ranges from 0 to  –The azimuth, , ranges from 0 to 2 

16 Quantization The Born interpretation puts restrictions on the acceptability of the wavefunction: 1.  must be finite –    2.  must be single-valued at each point 3.  must be continuous 4. Its first derivative (its slope) must be continuous These requirements lead to severe restrictions on acceptable solutions to the Schrödinger equation A particle may possess only certain energies, for otherwise its wavefunction would be physically impossible The energy of the particle is quantized

17 Solutions to the Schrödinger equation The Schrödinger equation for a particle of mass m free to move along the x-axis with zero potential energy is: (-h 2 /2m) d 2  /dx 2 = E  –V(x) =0 –h = h/2  Solutions of the equation have the form:  = A e ikx + B e -ikx –A and B are constants –E = k 2 h 2 /2m h = h/2 

Chapter 1118 The Probability Density  = A e ikx + B e -ikx 1. Assume B=0  = A e ikx |  | 2 =  *  = |A| 2 –The probability density is constant (independent of x) –Equal probability of finding the particle at each point along x-axis 2. Assume A=0 |  | 2 = |B| 2 3. Assume A = B |  | 2 = 4|A| 2 cos 2 kx –The probability density periodically varies between 0 and 4|A| 2 –Locations where |  | 2 = 0 corresponds to nodes – nodal points

19 Eigenvalues and Eigenfunctions The Schrödinger equation is an eigenvalue equation An eigenvalue equation has the form: (Operator)(function) = (Constant factor)  (same function)  =  –  is the eigenvalue of the operator  –the function  is called an eigenfunction –  is different for each eigenvalue In the Schrödinger equation, the wavefunctions are the eigenfunctions of the hamiltonian operator, and the corresponding eigenvalues are the allowed energies

20 Superpositions and Expectation Values When the wave function of a particle is not an eigenfunction of an operator, the property to which the operator corresponds does not have a definite value For example, the wavefunction  = 2A coskx is not an eigenfunction of the linear momentum operator This wavefunction can be written as a linear combination of two wavefunctions with definite eigenvalues, kh and -kh –  = 2A coskx = A e ikx + A e -ikx –h = h/2  The particle will always have a linear momentum of magnitude kh (kh or –kh) The same interpretation applies for any wavefunction written as a linear combination or superposition of wavefunctions

21 Quantum Mechanical Rules The following rules apply for a wavefunction, , that can be written as a linear combination of eigenfunctions of an operator  = c 1  1 + c 2  2 + …….. =  c k  k –c 1, c 2, …. are numerical coefficients –  1,  2, ……. are eigenfunctions with different eigenvalues 1. When the momentum (or other observable) is measured in a single observation, one of the eigenvalues corresponding to the  k that contribute to the superposition will be found 2. The probability of measuring a particular eigenvalue in a series of observations is proportional to the square modulus, |c k | 2, of the corresponding coefficient in the linear combination

Chapter 1122 Quantum Mechanical Rules 3. The average value of a large number of observations is given by the expectation value, , of the operator  corresponding to the observable of interest The expectation value of an operator  is defined as:  =   *  d  –the formula is valid for normalized wavefunctions

23 Orthogonal Wavefunctions Wave functions  i and  j are orthogonal if   i *  j d  = 0 Eigenfunctions corresponding to different eigenvalues of the same operator are orthogonal

24 The Uncertainty Principle It is impossible to specify simultaneously with arbitrary precision both the momentum and position of a particle (The Heisenberg Uncertainty Principle) –If the momentum is specified precisely, then it is impossible to predict the location of the particle By superimposing a large number of wavefunctions it is possible to accurately know the position of the particle (the resulting wave function has a sharp, narrow spike) –Each wavefunction has its own linear momentum. –Information about the linear momentum is lost

25 The Uncertainty Principle -A Quantitative Version  p  q  ½h –  p = uncertainty in linear momentum –  q = uncertainty in position –h = h/2  `Heisenberg’s Uncertainty Principle applies to any pair of complementary observables Two observables are complementary if  1  2   2  1 –The two operators do not commute (the effect of the two operators depends on their order)