General Physical Chemistry II Lecture 3 Aleksey Kocherzhenko September 2, 2014"
Last time "
The time-independent Schrödinger equation" Erwin Schrödinger " ~ 2 2m d 2 (x) dx 2 The wavefunction:" (x) The Hamiltonian operator:" Ĥ= ~2 2m Kinetic energy operator" + V (x) (x) =E (x) d 2 dx 2 + V (x) Potential energy operator" Time-independent Schrödinger equation (TISE) in operator form: " We also generalized the TISE to 3D case:" where," Ĥ= ~2 2m r2 + V (x, y, z) Ĥ (x) =E (x) Eigenfunction" Eigenvalue" Ĥ (x, y, z) =E (x, y, z),
The Born interpretation of the wavefunction" The probability of finding a particle in a small region" of space of volume V is proportional to " 2 V = V The wavefunction must be normalized:" ZZZ Max Born" (x, y, z) 2 dv =1 V The second derivative must exist, so the wavefunction must be continuous and smooth! (1D) time-independent " Schrödinger equation: " ~ 2 2m d 2 (x) dx 2 + V (x) (x) =E (x)
Let s use the time-independent Schrödinger equation to study a free particle "
Free particle in 1D" A free particle does not experience any external forces/potentials:" Ĥ= ~2 2m d 2 dx 2 + V (x) {z } 0 = ~2 2m The time-independent Schrödinger equation is then: " ~ 2 2m d 2 (x) dx 2 = E (x) p 2mE (x) =sin(kx), k = ~ d 2 dx 2 Solution: where (try showing this)" Remember: the wavenumber was defined as, k =2 / (x) =sin(kx) so the momentum of a free particle with wavefunction is " p = h/ = ~k = p 2mE
The position of a free particle" (x) =sin(kx) Momentum:" p = h/ = ~k = p 2mE But what is the position of a free particle?" (x) 2 The particle can be anywhere in space:" The position is not well defined!" 0 2 x x 2 ( 1, +1)
The momentum of a localized particle" (x) (x) = (x) 0 x But what is the momentum of a localized particle?" The delta function can be decomposed into a Fourier series: " (x) = 1 2 + 1 1X n=1 sin nx + 2 n =1, 2, 3,... Infinite number of modes:, each with its own wavenumber (and momentum) value!" (Approximating the delta function with a finite number of modes, )" n =2, 5, 21 The momentum is not well defined!"
Learning more about special functions" If you are interested, you can read more about the delta function, e.g., at" http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/fourierseries.pdf" or http://mathworld.wolfram.com/deltafunction.html (optional)"
The position-momentum uncertainty principle" It is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particle.! Mathematically:" x p ~ 2 x position uncertainty" Werner Heisenberg" p momentum uncertainty" The position-momentum uncertainty principle is just one (really important) manifestation" of Heisenberg s uncertainty principle"
Understanding the uncertainty principle "
Operators represent physical quantities" The Hamiltonian operator (total energy):" Ĥ= ~2 2m d 2 dx 2 + V (x) (1D)" Ĥ= ~2 2m r2 + V (x, y, z) (3D)" Kinetic energy operator" The momentum operator (1D):" ˆp x = Potential energy operator" i~ d dx (Let s show this)" The momentum operator (3D):" ˆp = i~ ~ r The position operator: ˆx = x (1D); ˆr = ~e x x + ~e y y + ~e z z (3D)"
Linear operators" All operators that are used in (traditional) quantum mechanics are linear operators:" Â[c 1 f 1 (x)+c 2 f 2 (x)] = c 1 Âf 1 (x)+c 2 Âf 2 (x) Exercise :" " Is the operator Â, defined by the expressions below, linear?" 1)" 2)" 3)" Âf (x) = p f (x) Âf (x) =x 2 f (x) Âf (x) = dn dx n f (x)
Measurement" It is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particle.! Ø To specify the value of a physical quantity, we need to measure it." Ø A measurement changes the state of the system!" Ø The state of macroscopic objects usually changes little due to measurement." (Example: thermal equilibration of thermometer and object of T measurement.)" Ø The state of microscopic objects may change significantly." Ø Furthermore, for microscopic objects, allowed values of physical quantities are quantized " Ø Measurement may lead to transition between quantum states!" In quantum mechanics, the terms measurement and state preparation are synonymous.!
Mathematically describing measurement" Ŝ s s For an operator, corresponding to physical quantity, the values of that we might find as a result of a measurement are given by eigenvalue equation:" Ŝ n (x) =s n n (x) s n n th allowed value of s (an eigenvalue) " n (x) wavefunction in the state corresponding to this eigenvalue" (x) It can be shown that the wavefunction of any state that a particle may be in can be written as a linear combination of eigenstates : " (x) = X n c n n (x) n (x) s When a measurement of is performed, we do not know what result we will get, but the probability of getting the result s n is given by" c n 2 = c nc n
Schrödinger s cat" Cat is alive inside a black box." A random process, e.g. radioactive decay, activates a hammer that breaks a vial with poison. Cat dies." A still from A Serious Man, by Ethan and Joel Coen" Initial wavefunction:" cat = c 1 cat alive + c 2 cat dead Measure (open black box and check whether cat is alive or dead):" Final wavefunction: cat alive probability " cat dead c 1 2 probability c " 2 2 Most likely, the cat was never both alive and dead (initial wf wrong). But particles can be in a superposition of two states!" (Also, we might have incorrectly defined measurement.)"
Measurement of two physical quantities" For two physical quantities, s and q è " Initially, the particle is in state" measure"s (x) = X n c n n (x) s : Ŝ n (x) =s n n (x) q : ˆQ n (x) =q n n (x) some value of "n s m, 0 (x) = m (x) m (x) = X n c 0 n n (x) measure"q q m 0, 00 (x) = m 0 (x) m 0 (x) = X c 00 n n (x) measure s s (again)" m 00, 000 (x) = m 00 (x) n In general, m (x) 6= m 0 (x) and m 6= m 00, so" m (x) 6= m 00 (x) Ø We only know the value of a physical quantity after we have measured it" Ø Measuring a second physical quantity may change the state of the particle, so we will no longer know the value of the first quantity that we measured"
Commuting operators" For two physical quantities, s and q è " s : Ŝ n (x) =s n n (x) q : ˆQ n (x) =q n n (x) Ŝ ˆQ f (x) Ŝ ˆQf (x) = ˆQŜf (x) The operators and are said to commute if for any (allowed) function, " ˆx = x ˆp x = i~ d dx 6= i~ d [xf (x)] = i~ dx Example: the operators and do not commute" apple x i~ df (x) dx ˆx = x ŷ = y xyf (x, y) =yxf (x, y) Example: the operators and do commute" apple df (x) f (x)+x dx It can be shown that if operators Ŝ and ˆQ commute, " n (x) = n (x)
Measurement with commuting operators" For two physical quantities, s and q è " Initially, the particle is in state" measure"s (x) = X n c n n (x) s : Ŝ n (x) =s n n (x) q : ˆQ n (x) =q n n (x) Assume that Ŝ and ˆQ commute è " n (x) = n (x) m (x) =1 m (x)+ X n6=m 0 n (x) measure"q s ˆQ some value of "n s m, 0 (x) = m (x) q m, 00 (x) = q Measuring did not change the state of the particle!" m (x) = m (x) Measuring again will not change the state of the particle either (show this)! " Ŝ Ø If and commute, then we can know the values of corresponding physical quantities and q at the same time with arbitrary precision!" s
Heisenberg s uncertainty principle" For two physical quantities, s and q è " Ŝ ˆQ s : Ŝ n (x) =s n n (x) q : ˆQ n (x) =q n n (x) Ø If and commute, then we can know the values of corresponding physical quantities and q at the same time with arbitrary precision." Ŝ ˆQ s Ø If and do not commute, then we cannot know the values of corresponding physical quantities s and q at the same time with arbitrary precision:" ~ s q 2 ( s and q are the uncertainties of s and q values)" Ŝ ˆQ n (x) = n (x) Ø This is because (unless and commute, and thus ), measuring the value of s changes the value of q in an unpredictable way, and vice versa! Ø The position-momentum uncertainty principle is a special case of Heisenberg s uncertainty principle for non-commuting operators ˆx and" ˆp x
A philosophical debate" Not everyone appreciated the probabilistic interpretation" of quantum theory " God does not play dice with the universe! " (Einstein believed that quantum theory is incomplete, and that by introducing hidden parameters it could be made deterministic)" Albert Einstein" Einstein, don t tell God what to do! " Niels Bohr" Quantum theory is remarkably successful at explaining experiments" There were attempts to build theories with hidden parameters, e.g., by David Bohm, but they gave no advantage over standard quantum theory" At present, most people accept that quantum theory is probabilistic"
Summarizing what we have learned "
Some postulates of quantum mechanics" 1) The state of a quantum-mechanical system is completely defined by its wavefunction (x). All dynamical information about the system can be derived from the wavefunction. (x) (x)dx gives the probability of finding the particle within a small interval from position." s 2) To every physical quantity that can be measured in classical mechanics (often referred to as an observable) corresponds a linear operator Ŝ in quantum mechanics." 3) In any measurement of the observable associated with operator, the only values that will ever be observed are the eigenvalues s n, found from the eigenvalue equation:" s dx Ŝ (x) =s n (x) 4) If a system is in a state described by a normalized wavefunction, then the average value of the observable corresponding to Ŝ is given by" +1 hsi = Z 1 (x) Ŝ (x)dx x Ŝ (x)
Average values from the wavefunction" Ŝ Ø For an operator that corresponds to physical quantity (observable)," +1 hsi = Z (x) Ŝ (x)dx s Example:" ˆx = x hxi = Z+1 1 (x) x (x)dx = Z +1 x (x) 2 dx 1 Another example:" ˆp x = i~ d dx Z hp x i = i~ +1 1 1 (x) d (x) dx dx
Let s start really studying quantum systems!"
Applications of quantum theory in chemistry" To find the properties of molecular systems based on quantum mechanics, we need to solve the time-independent Schrödinger equation with the appropriate Hamiltonian" Three types of motion:" Ø Translation" z Ø Vibration" z Ø Rotation" z y x x y x y These types of motion play an important role in chemistry, because they are the way in which molecules store energy"
Model systems" We are going to consider four common model systems:" Ø Particle in a box " "Translational motion (electron movement in polyenes)" Ø Particle on a ring " "Rotational motion (electron movement in cyclic molecule)" Ø Particle on a sphere " "Rotational motion (electrons rotating around the nucleus)" Ø Harmonic oscillator " "Vibrational motion of diatomic molecules" These types of motion play an important role in chemistry, because they are the way in which molecules store energy"
Summary" Ø Physical quantities (observables)" are represented in quantum mechanics by linear operators! Ø The physical basis for Heisenberg s uncertainty principle! is that measurement changes the state of the system! s Ø For an observable that is described by the operator, the only values s n that might appear in a measurement" are those that satisfy the eigenvalue equation:! Ŝ n (x) =s n n (x) Ø Only values of observables that correspond to commuting operators (and thus, have common eigenfunctions)" can be known simultaneously to arbitrary precision! Ŝ