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Unformatted text preview: Chemistry 260/261 Today in Chemistry 260/261 The Fourier Theorem and quantum mechanical operators Hamiltonian operator and Schrodinger’s Equation meaning of the wave function (Max Born) particle on a line (1D Particle in a Box) •
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• This Week in Chemistry 260/261 •
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• Reading: (Fri) 4.7; 5.1 169 174. Harmonic Oscillator 2 and 3 dimensional particle in a box Hydrogen atom Problem set 2 due Friday, September 16 by 3:00 pm Lecture 4
January 12, 2010 Wave/Particle Duality Waves give the distribution, but often particles are detected one by one Simulation of Young’s double slit experiment, one particle at a time: Each electron makes a single spot on the Wilm The sum of many single spots produces an interference pattern identical to that for light For each electron, we know where it can go, but not where it will go. Double slit experiments – a shock introduction to QM ANIMATION: Check out: “Dr quantum double slit” On youtube States: A. only (a) open B. only (b) open C. both (ab) open. Regimes of (ab) event: 1. Classical particles – Slow emission, No interaction 2. Classical wave – Diffraction pattern 3. microscopic system (QM) – Short times one by one emission – Random frame NOT wave Long times – Fringe develop  NOT particle è༎ The appearance of interference builds up one quantum particle at a time è༎ As long as the ‘slit’ is comparable to the size of the quantum object 1 Another diffraction experiment: The Airy Pattern Another diffraction experiment: The Airy Pattern Light or electrons passing through a circular aperture is diffracted and forms a pattern of light and dark regions on a screen some distance L away from the aperture. θ = 70( L / d ) Light or electrons passing through a circular aperture is diffracted and forms a pattern of light and dark regions on a screen some distance L away from the aperture. θ = 70( L / d ) First ring occurs at (Const* L/d) First ring occurs at (Const* L/d) When d is small (small uncertainty in position)… the Airy pattern has a large diameter. When d is large (great uncertainty in position) … the Airy pattern has a small diameter. When d is small (small uncertainty in position)… the Airy pattern has a large diameter. Small uncertainty in position – large uncertainty in momentum. Large uncertainty in position – small uncertainty in momentum. Orders of magnitude (or when should we care about ‘small’?) Electron with velocity 1×106 ms 1 Baseball of mass 0.145 kg with velocity 30 ms 1 h
h
⇒λ= =
= 0.73 nm
p me v
atomic size! h
h
⇒λ= =
= 1.5 × 10−34 m = 1.5 × 10−25 nm
p mv
too small to be observed with any known experiment! 7 cm = 0.07 m The Indeterminacy Principle Like it or not, all experiments tell us that we can only describe quantum systems probabilistically. quantitative form: 1
ΔxΔp x ≥ 2 Δp Δx
position = h
2π … where x and p are along the same axis momentum The uncertainty principle matters when the wavelength is comparable to the certainty in position Prob. position momentum
Prob. The wavelength of a baseball is inWinitesimal compared to its size. If you throw a ball through a hole the diffraction is insigniWicant. Prob. The wavelength of an electron is much larger than its size. If you shoot an electron through a small hole it will diffract. Prob. Heisenberg showed that for certain quantities, such as position and momentum, the two probability distributions are not independent. position momentum 2 Birth of the Quantum Theory Facing the (quantum) facts A “quantum theory” needed three things: Classical physics was a world of matter and 5ields (i.e. particles and waves). Matter was particulate; light was wave like. 1) some mathematical quantity that stands for quantum “stuff” 2) A law that describes how this quantum stuff goes through changes • Planck reasoned that energies must be quantized in order to make a reliable physical model of blackbody radiation. • Einstein and Compton showed that light (electromagnetic radiation) was particle like (photoelectric effect, Compton scattering). • deBroglie predicted that matter particles have wave properties 3) A rule of correspondence that tells how to translate the theory’s symbols into activities in the world Three (!) quantum theories were put forward around 1925… 1) matrix mechanics (Werner Heisenberg) • each matrix represents a different observable (e.g. momentum, position) • matricies followed a law of motion similar to Newton’s laws (with notable differences!) • Davisson and Germer later proved it by diffracting electrons. • Heisenberg noted a strange experimental relationship between pairs of experimental observables… things were interrelated via probability. The wave particle distinction of classical physics was forever dissolved… more for “5ields” or “matter” … everything was quantum “stuff”. 2) wave mechanics (Erwin Schrödinger) •
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• represents quantum “stuff” as a wave form wrote quantum laws of motion (Schrödinger equation) Wave imagery is very dubious… these aren’t classical waves! 3) transformation theory (Paul Dirac) What was this quantum “stuff” and how could we model it? • The wave function connects wave and matrix mechanics Waves on a string An introduction to the classical wave equation and wave like behavior Wave mechanics: a particle is spread through space like a wave… precise location of particles at any time f(x) 0 We can no longer use the classical trajectory trajectory 0 x The motion of a string is governed by the classical wave equation L 2 d f ( x)
= −k 2 f ( x)
dx 2 We must use wave functions describes where the particle is most likely to be found (Ψ2 is the probability) wave function wave function Ψ is the wave function  the modern term for de Broglie’s “matter wave” Contains all the information one would ever want to know about the quantum mechanical system. Wave mechanics concerns itself with calculating and interpreting Ψ Take Wirst derivative: We know two kinds of functions whose derivatives look like the function itself: sine, cosine and exp. Let’s try sine and cosine in the most general way: f ( x) = A sin kx + B cos kx d
d
f ( x) = ( A sin kx + B cos kx ) = Ak cos kx − Bk sin kx
dx
dx
Take second derivative: d2
d
f ( x) = ( Ak cos kx − Bk sin kx ) = − Ak 2 sin kx − Bk 2 cos kx
dx 2
dx
Hey, that worked! We get − Ak 2 sin kx − Bk 2 cos kx = −k 2 ( A sin kx + B cos kx ) our function back times a negative constant. 3 Waves on a String: Boundary Conditions f ( x) = A sin kx + B cos kx f(x) 0 = 0 f(0) 0 x Waves on a String: Boundary Conditions f(L) = 0 L The wave is conWined to 0 at the ends, we have boundary conditions Now we have a function that solves the wave equation, but we have not used all the information yet. We know k, but not A and B. f (0) = 0; f ( x) = A sin kx + B cos kx f(x) 0 = 0 f(0) 0 x f(L) = 0 L The wave is conWined to 0 at the ends, we have boundary conditions f ( L) = 0 First case, x=0: Second case, x=L: f ( x = 0) = A sin k 0 + B cos k 0 = 0 + B ×1 = B Whenever we have kL=nπ for n=1,2,3,4… The wave function will be zero for x = L. k= The Wave Function  Born Interpretation wave function Born made an analogy to the wave theory of light: ψ ( x )d x
x2 ∫ x1 ∫ −∞ ∫ −∞ ∞ ∞ 2 nπ
, n = 1, 2, 3, ...
L No assumptions, besides boundary conditions! What makes a good wave function? Max Born gave the wave function physical meaning… 2 f ( L) = 0 f ( x = L) = A sin kL = 0 f ( x) = A sin kx + 0 cos kx = A sin kx ψ 2(x) f (0) = 0; We can divide by A, leaving: sin kL = 0 Oh no! This is bad, we need the function to be zero. So that means B must equal zero. de Broglie gave us the “matter wave” – the wave function Ψ Now we have a function that solves the wave equation, but we have not used all the information yet. We know k, but not A and B. probability distribution A wave function MUST be: 1. single valued (i.e. a wave function must have only one possible f(x) value for each and every value of x) 2. continuous 3. differentiable (e.g., there must be no mathematical reasons why the derivative of Ψ cannot exist)... Ψ must
be bounded probability of Winding a particle between x and x + dx ψ ( x )d x probability of Winding particle between x1 and x2 ψ 2 ( x )d x probability of Winding particle ψ 2 ( x )d x = 1 NOT SINGLEVALUED normalization requirement The probability of Winding a particle in a small region of space of volume dV is proportional to Ψ2dV. NOT BOUNDED
x x NOT CONTINUOUS
x x 4 Operators Follow the energy! de Broglie: a particle moving in 1D with momentum p is described by the following wave function: ! 2π x $
h
ψ ( x ) = A sin #
&, where λ =
"λ%
p 2×3= 6 If we take the 2nd derivative of our wave function: 2 operator 2 " 2π %
" 2π p %
d2
ψ ( x) = −$ ' ψ ( x) = −$
' ψ (x)
#λ&
#h&
dx 2
2 2 2 2 4π p h
p
1
=
= mv 2
h 2 8 mπ 2 2 m 2
h2 d 2
ψ ( x ) = EK ψ ( x )
8mπ 2 dx 2 2 d 2 f ( x)
= −k 2 f ( x)
dx 2
2 2 4π h p
⋅
4π 2 h 2 2m We should include both potential and kinetic energy: 2 d 2
−
ψ( x) + V ( x)ψ( x) = ETotal ψ( x)
2m dx 2 An operator is a mathematical instruction: “Do something to this function or these numbers”. Multiply by one and rearrange = h
2π This is easier to deal with if we rewrite it in a form of an operator operating on a function. … Huh? ˆ
M (2, 3) = 6 f(x) d
( 3x 3 + 4 x 2 + 5 ) = 9 x 2 + 8 x
dx ˆ
D[ f ( x )] = 9 x 2 + 8 x operator # −h2 d 2
ˆ&
% 8π 2 m dx 2 + V ( Ψ
$
'
operator ˆ
HΨ
Hamiltonian Operator Total Energy Operator Follow the energy!: the Hamiltonian operator Operators When an operator acts on a function, some other function is usually generated. The Hamiltonian operator including both kinetic and potential energy: Special type of operator/function combination: d2
d
(sin 4 x ) = ( 4 cos 4 x ) = −16 sin 4 x
2
dx
dx
…you get a constant times when you evaluate the function… In operator symbolism: the original function. ˆ
BΨ = K Ψ Ψ is an eigenfunction of the operator this is known as an eigenvalue equation the constant K is an eigenvalue From the consideration of the de Broglie wave: h2 d 2
ψ ( x ) = EK ψ ( x )
8mπ 2 dx 2 We get an eigenfunction eigenvalue expression that relates the 2nd derivative of the wave function to kinetic energy of a free particle! ⎡ྎ h 2 d 2
⎤ྏ
+ V ( x)⎥ྏ ψ( x) = Eψ( x)
⎢ྎ− 2
2
⎣ྏ 8π m dx
⎦ྏ ˆ
Hψ = Eψ
Schrödinger’s Equation An eigenvalue equation Not all functions are eigenfunctions of all operators… this is a rare occurrence on which much of wave mechanics is built. The eigenfunctions of the Hamiltonian are particularly important. à༎ These states – these wave functions – describe matter when energy is conserved. à༎ Energy “eigenstates” are the matter wave equivalent to standing waves à༎ Orbitals are energy eigenstates for the electron – 1s, 2s, 2p, … 5 The Wave Function Are you following the energy??? Quantum Mechanics Classical Mechanics h2 d 2
h2 d 2
ˆ
K=
=−
2
2
8mπ dx
2m dx 2 1 2 p2
K = mvx = x
2
2m
q1 ⋅ q2
V=
4πε 0 r (1 D) ˆ q ⋅q
V= 1 2
4πε 0 r Contains all the information one would ever want to know about the quantum mechanical system. HUH? The Fourier Theorem The Fourier theorem states that a periodic function, f(x), which is reasonably continuous, may be expressed as the sum of a series of sine and cosine terms. Of Note… The Schrödinger equation and the procedures for calculating values from it, can be deduced – but they cannot be derived. They have the same status as Newton’s laws and Maxwell’s equations  these are all laws of nature, or postulates, whose only justiWication comes from their consistency with experiment. The behavior of electrons is described by a wave function. The wave function is used to determine all properties of the electrons. Values can be predicted by operating on the wave function with the appropriate operator. The appropriate operator for predicting energy of the electron is the Hamiltonian Operator. http://phet.colorado.edu/ Sine waves form a universal alphabet in terms of which any wave can be written. In wave mechanics, this “alphabet” has special meaning. “Filtering” the wave function How can we access all the information a wave function has to offer? We want to know information concerning observables (attributes): 1. Position 2. Momentum 3. Energy 4. Spin (magnitude/orientation) As it turns out, each attribute we want to know about is associated with a wave form…
one of quantum theories more bizarre features: 1. position… impulse wave 2. momentum… spatial sine wave 3. energy… temporal sine wave 4. spin (magnitude/orientation)… spherical harmonics Any wave function can be written as a combination of the above families For the Fourier simulator: http://phet.colorado.edu/en/simulation/fourier We can access all this information by applying Fourier’s theorem to our wave function. We can “5ilter” our wave function by using an OPERATOR. 6 Example: 1D Particle in a box Example: 1D Particle in a box A particle is conWined in a region of space in one dimension, x. V(x) ∞ The potential function is: ∞ 0 e– ∞ 0, if 0 < x < L ∞, otherwise V= x L V(x) d2
ψ ( x ) = − k 2ψ ( x )
dx 2 ∞ The wave equation problem all over again… 0 e– x L ψ ( x ) = A sin kx + B cos kx We can write the Schrödinger equation: − Need a function that after you take two derivatives gives you itself back times a constant… Since m, E and h are all positive, we know that the constant must be negative (from the minus sign). 2 d 2
ψ ( x ) + Vψ ( x ) = Eψ ( x )
2 m dx 2
2 d 2
ψ ( x ) = Eψ ( x )
2 m dx 2 d2
2 mE
ψ ( x ) = − 2 ψ ( x ) = − k 2ψ ( x )
dx 2 ∞ 0 e– L x Since it is related to probability, and we know the electron cannot be outside the box, we have boundary conditions First case, x=0: Second case, x=L: d2
d
ψ ( x ) = ( Ak cos kx − Bk sin kx ) = − Ak 2 sin kx − Bk 2 cos kx
dx 2
dx − Ak 2 sin kx − Bk 2 cos kx = − k 2 ( A sin kx + B cos kx ) our function back times a negative constant. Example: 1D Particle in a box ψ ( x = 0, x = L ) = 0 ψ ( x = L ) = A sin kL = 0 kL = nπ , n = 1, 2, 3, ... No assumptions, besides boundary conditions! nπ x
, n = 1, 2, 3, ...
L Already, we can calculate the energy: ∞ ∞ 0 e– L k2 =
x 2 mE " nπ %
=$ '
2 # L & 9h2
8 mL2 2 2 2 Look at energy level trends: E ∝ n2 4h2
8 mL2
2 h
8 mL2 2 "h%
n2 $ ' π 2
# 2π &
nπ
n2h2
E=
=
=
, n = 1, 2, 3, ...
2 mL2
2 mL2
8 mL2
2 The same boundary condition we had for a string. 0 ψ ( x ) = A sin V(x) Now we have a function that solves the Schrödinger equation, but we have not used all the information yet. We know k, but not A and B. What else do we know about the wave function? ψ ( x = 0 ) = A sin k 0 + B cos k 0 = 0 + B × 1 = B nπ
k=
, n = 1, 2, 3, ...
L Take second derivative: 2 mE
2 ψ ( x ) = A sin kx + B cos kx ∞ d
d
ψ ( x ) = ( A sin kx + B cos kx ) = Ak cos kx − Bk sin kx
dx
dx Hey, that worked! We get k2 = Example: 1D Particle in a box: Boundary Conditions V(x) Take Wirst derivative: energy − E∝ 1
L2 Look at differences between levels: ΔEnm = En − Em = n2h2 m2h2
h2
−
=
n2 − m2
2
2
8 mL 8 mL
8 mL2 ( ) 7 ψ ( x ) = A sin V(x) nπ x
, n = 1, 2, 3, ...
L Example: 1D Particle in a box ψ ( x) =
V(x) What about A? Born: the square of the wave function is the probability, so if we add up the probability over the whole box, we had better 5ind the electron in there! ∞ x L L nπ x
∫ ψ ( x ) dx = A ∫ sin L dx = 1
0
0
2 2 We call this NORMALIZATION. L energy 2
∫ sin
0 0 nπ x
# 1 sin 2 nπ &
dx = L % −
(
$2
L
4 nπ ' L 4h2
8 mL2 nπx
⎛ྎ L ⎞ྏ
A ∫ sin
dx = A2 ⎜ྎ ⎟ྏ = 1
L
⎝ྎ 2 ⎠ྏ
0 h2
8 mL2 A2 = 2 Example: 1D Particle in a box 2 2
L ψ ( x) = A= 2
nπ x
sin
L
L n=2 x e– 0 2
n probability density ψ ( x) L x Prob(x1 < x < x2) n=2 x n2h2
, n = 1, 2, 3, ...
8 mL2 Summary: What we know from solving the particle in a box problem So What! What does this have to do with anything real? V x2 0 x1 x ψ(x)2 V(x) = 0 V(x) = ∞ = ∫ ψ 2 ( x)dx 2 This is the probability distribution ψ(x) − V(x) = ∞ L ψ n (x) = E α n2 d ψ ( x) ˆ + V ( x ) ψ ( x ) = Eψ ( x )
2m dx 2 ψ(x)2 16 2 d 2ψ ( x )
−
= Eψ ( x )
2 m dx 2 ψ ( x ) = A sin( kx )
wavefunctions BOUNDARY CONDITIONS 2
# nπ x &
sin %
$L(
'
L x 2 n=1 x x x Look at the distributions. We can calculate the probability of Winding the electron between two points as: ∞ x n=2 n=1 n=1 V(x) ∞ ψ ( x) x 2
L En = n2h2
, n = 1, 2, 3, ...
8 mL2 2
n n=3 ψn ( x ) En = n=3 x L L 2 9h2
8 mL2 e– 2
nπ x
sin
L
L Now we know everything! We can look at the wave functions: ∞ 0 Wave function e– 0 ∞ probability density ∞ E= n2 h2
8mL2 Energy Example: 1D Particle in a box 9 4 quantized energies 1 8 Relevance to chemistry 1 2 A (λ ) 3 Chlorophyll a λ
ΔEnm h2
= En − Em =
n2 − m2
8mL2 ( Chlorophyll b Carotenoid ) 4 Structures from wikipedia 9 ...
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This note was uploaded on 01/19/2012 for the course CHEM 260 taught by Professor Staff during the Fall '08 term at University of Michigan.
 Fall '08
 STAFF
 Chemistry

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