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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
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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 ( cv,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
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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
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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
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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 ho = is the work function (essentially equal to the ionization potential of
the metal)
• The kinetic energy of the ejected particle is given by:
• ½mv2 = h - • The photoelectric effect shows that the incident radiation is
composed of photons that have energy proportional to the frequency of the radiation
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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 11 9
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):
• (-h2/2m) d2/dx2 + 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 11 10
The Schrödinger Equation
• The Schrödinger equation for a particle moving in three dimensions can be written:
• (-h2/2m) 2 + V = E 2 = 2/x2 + 2/y2 + 2/z2
• The Schrödinger equation is often written:
• H = E– H is the hamiltonian operator
– H = -h2/2m 2 + V
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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 2dx 2 is the probability density
– is called the probability amplitude
Chapter 11 12
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 2d– d = dx dy dz
– d is an infinitesimal volume element
• P 2 d– P is the probability
Chapter 11 13
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)
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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
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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 = r2 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
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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
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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:
• (-h2/2m) d2/dx2 = E– V(x) =0
– h = h/2• Solutions of the equation have the form: = A eikx + B e-ikx
– A and B are constants
– E = k2h2/2m• h = h/2
Chapter 11 18
The Probability Density
= A eikx + B e-ikx
• 1. Assume B=0• = A eikx
• ||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 cos2kx
– The probability density periodically varies between 0 and 4|A|2
– Locations where ||2 = 0 corresponds to nodes – nodal points
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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
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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 eikx + 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
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Quantum Mechanical Rules
• The following rules apply for a wavefunction, , that can be written as a linear combination of eigenfunctions of an operator
= c11 + c22 + …….. = ckk
– c1 , c2 , …. 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, |ck|2, of the corresponding coefficient in the linear combination
Chapter 11 22
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
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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
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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
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The Uncertainty Principle -A Quantitative Version
pq 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 12 21
– The two operators do not commute (the effect of the two operators depends on their order)