Quantum Mechanics Flashcards
Classical Mechanics
E=1/2mv² + V = P²/2m + V
EK - due to momentum (P)
V - due to position (x)
Need to know starting position, P and V to predict the subsequent trajectory exactly.
Newtonian mechanics - any x and p (or E) is possible.
Rotations of bodies
p->J/L (angular momentum) m-> I (moment of inertia) v -> ω (angular velocity) F -> T (torque) Any rotational E/p possible, if suitable torque is applied
Vibrations of bodies
F=-kx
All E possible
Internal motion
Light
Classical view - light is a wave (reflection, refraction etc)
Modern view - wave-particle duality
EMR-light has different colours associated with a Δλ.
Continuous, all E and λ possible.
Blackbody radiation
Material that converts heat to light. Cavity with exit pin-hole for light to escape, emitted by hot body. Increase T results in increase in total output, shifting the maxima to a lower λ.
ρ = total E in region of EM field / V
- ρ is energy density
Wein’s law
Assumptions about absorption/emission of EMR.
ρ=c1/λ⁵.1/e(c2/λT), adjusting c1 and c2 to get best fit.
-really good at short λ
-not so good at long λ
Rayleigh Jean Law
Treat EM field as a lot of oscillators. Calculate the number with each E, predict a curve proportional to λ-⁴
Full and detailed thermodynamic treatment.
ρ=8πkT/λ⁴
-good at long λ
-terrible at short λ
Average of each oscillator is kT, regardless of frequency. Each wave mode has the same energy.
UV catastrophe - emissions due to ∞ at 0λ. High E’s even at low T (no darkness!). All bodies lose E fast - does not reflect reality.
Planck Distribution
Empirical modification of Wein’s law.
ρ=c1/λ⁵.1/e(c2/λT)-1
Essentially a perfect fit at all λ.
Each mode has a discrete amount of E (QM). Addition of the -1 term, so cannot have continuous values (as previously allowed)
Quantum Assumptions
Wall of blackbody radiator contains harmonic oscillator (HO), each with characteristic frequency. E of each HO quantised: E=nhv, where n=0,1,2... HOs emit and absorb E only in discrete amounts: E=hv. Assumptions: c1=8πch c2=ch/k(B) ρ=8πch/λ⁵.1/e(ch/λKT)-1
Photoelectric Effect
Light is a particle.
When light is shone on some metals, electrons are emitted. EK of e-s remains the same. No e-s emitted for v
e- diffraction
e-s diffracted from metallic Ni. Gives a diffraction pattern exactly in accord with de Broglie’s λ. Cannot think of small objects in terms of trajectory (x and p) - Heisenberg uncertainty. Rather, a probability of being in any particular place, described by the wavefunction.
Wavefunctions Ψ
Mathematical function describing the probability distribution of a particle (e-). Ψ is a function of position and time.
Deal with time-independent systems -> only position (x,y,z)
Postulate 1 - Ψ
In 1D Ψ, Ψ(x), allows us to determine translational motion.
Ψ contains all the dynamical information about the system. Values have no physical meaning. Get information by manipulating Ψ to return real values relating to physical properties (E, x, p…).
Operator form of SWE: H^Ψ=EΨ
Postulate 2 - Born Interpretation
Wavefunction has value Ψ at some point r. Probability of finding the particle in a really small volume at r∝|Ψ²|dτ
|Ψ²| gives real, positive values (relating to where something is)
Postulate 3 - acceptable Ψ’s
- continuous
- single values
- continuous first derivative
- square integrate the function (not infinite)
Operators
Something you do to a function to get real information out. Different operators represent different properties (E, p, x…)
Postulate 4 - observables
Represented by operators (built from x or p operators).
-position operator (x^) is x multiplied by something (operator for position along x axis). Multiply Ψ by x.
-momentum operator (p^) = ℏ/i . d/dx
Satisfy commutation relation (relation between conjugate quantities - mutually exclusive). x^ and p^ in x-direction of a particle in 1D: x^1p^ = xP-Px
-communtator of reduced Planck’s constant and i: x^p^=iℏ
Postulate 5 - Heisenberg
Impossible to specify simultaneously both p and x of particle. Can relate any pair of observables with operators that do not commutate.
ΔpΔx>ℏ/x
px-xp=h/2πi
If x=0, Δp=infinity - complete uncertainity in momentum
If p=0, Δx=infinity - complete uncertainity in position
Time independent SWE
Interpretation of de Broglie (wave-particle duality)
-ℏ²/2m . d²Ψ/dx² + V(x)Ψ = EΨ
Gives real values
Solutions: Ψ=coskx + isinkx
-k can take any value, thus E should be able to take any value. But acceptable Ψ restricts E (quantised)
K(B)
Boltzmann Constant
H^
Hamiltonian Operator
τ (tau)
Really small volume
dτ=dxdydz
|Ψ²
Probability density
i
Imaginary number = root(-1)
ℏ
Reduced Planck’s constant = h/2π (Js/rad)
Simple model systems
Translational (movement in 1D), rotational (motion about an axis) and vibrational (internal motion).
Translational motion
Description of motion in 1D
Particle in a box and at a barrier
Particle in a box
Particle of mass m can lie anywhere along a line of length L, creating a box where you can find the e- (V=∞ at walls and V=0 at base). Probability of being where V=∞ is zero, Ψ=0 at x=0 and x=L.
-ℏ²/2m . d²Ψ/dx² = EΨ as V=0
General solutions: Ψ=Ae(ikx) + Be(-ikx)
E=ℏk²/2m describes Ψ and E of a free particle.
As the particle is confined to a region, the acceptable Ψ must satisfy boundary conditions.
Ψ=(A+B)coskx + (A-B)isinkx = Csin(kx) + Dcos(kx)
x=0, Ψ(0)=0.
-D=0, thus, Ψ = Csin(kx)
x=L, Ψ(L)=Csin(kx)=0.
-if C=0, then Ψ=0 for all values of x, conflicting with the Born Interpretation (particle is somewhere)
-sin(kL)=0 and kL=πn, n=1,2,3…
-Ψ(x)=Csin kπx/L = Csin nπx/L
E=n²h²/8mL²
C=root(L/2), thus, Ψ(x)=root(2/L)sin(nπx/L) for 0
Vibrating string
Classical analogy: only certain wavelengths possible as the ends are fixed.
Ψ = sin 2πx/λ, λ restricted to 2L/n, n=1,2,3…
-quantisation forced by boundary conditions
Energy levels - translational motion
E levels increase as n² increases, separation increases with n.
E(n)=n²h²/8mL², n=1,2,3…
Zero-point energy (ZPE)
n cannot = 0, so E cannot = 0.
n=1, E(1)=h²/8mL² = ZPE (lowest E a particle can possess).
-must have non-zero kinetic energy (Heisenberg)
-Ψ=0 at walls, but smooth, continuous and non-zero everywhere else.
Correspondence Principle
The behaviour of QM systems becomes more classical in the limit of large qns.
E(n+1)-E(n)=(2n+1)h²/8mL²
E will increase with decreasing m and L (L has a greater effect due to ²).
-As m tends to infinity, E tends to 0. Thus, E level separation becomes inseparable and is no longer quantised
Implications - atoms/molecules in beakers treated as if their translational motion is not quantised, impacting spectroscopy and material properties.
Applications - predict dye colour
To be coloured, dyes must absorb in the visible region of the EM spectrum. Interaction of light with e-s. Absorption of light/photons (change of quantised E). Object absorbing certain λ or v of white light, reflects complementary colour. E=hv=hc/λ
π e-s in p-orbitals (conjugated systems - alternating single and double bounds).
Terminantor atoms - denote the end of the box, disrupt the flow of e-s along molecule (N)
1) box length: 1.39Å x #bond lengths
2)#e-s (lp and π) - for multiple e-s, assume no interaction between e-s apart from Pauli Exclusion Principle. Populate using Aufbau’s rule.
3) ΔE=(2n+1)h²/8mL² (should be E-19 J or kgm²/s²). Convert to kJ/mol by multiply by N(A)/1000.
4) λ = hc/E. Absorbs this λ and reflects complementary colour
5) v = E/h
Particle at a Barrier
Tunnelling - particle penetrates through the walls to classically forbidden regions. Walls thin, enough E for e- to pass through.
E=ℏ²k²/2m
kℏ²=root(2mE)
V->infinity at barrier, but if it doesn’t, E A’e(ikx).
Particle has a probability of being beyond the barrier, and within the barrier. Ψ must be continuous and dΨ/dx must be continuous entering and leaving the barrier.
Tunnelling consequence of wave character behaviour of matter (ex: radio waves pass through walls). Matter can pass through walls on the quantum level. Probability decreases with increasing thickness of walls and m of particle (e->proton).
Applications of particle at a barrier - Scanning Tunnelling Microscopy (STM)
STM - dependence of e- tunnelling on thickness of region between point and surface. Needle scans across surface of conducting solid and e-s tunnelled across space.
Constant current mode - fixed distance between surface and stylus -> map
Quantum biology - does tunnelling occur during mutations of DNA? - Life and cancer research.
c=normalising constant
c=root(L/2)
Vibrational motion
Two models, Harmonic oscillator (HO) and Anharmonic oscillator (AO)
Harmonic Oscillator (HO)
Hooke’s law: F=-kx
K is force constant. Large K = narrow potential well = stiff spring (more F required to displace spring)
V=1/2kx² - parabolic shape. No displacement means particle is at eqm. Particle tends towards infinity, no sharp boundary (unlike particle in a box)
SWE: -ℏ²/2m . (1/2x²)Ψ = EΨ
-particle trapped by infinite walls, results in quantisation and boundary conditions. Ψ and Ψ² don’t equal 0 at any point (except infinity).
-particle can be anywhere, regardless of E (even chemically forbidden zone -> tunnelling).
E levels: E=(v+1/2)ℏω, where v=0,1,2… and ω=root(k/m)
E spacings are even (ΔE=ℏω). For large objects ℏω is negligible.
ZPE: lowest E level at 1/2ℏω (E cannot be 0 - always in motion and position restricted). Walls rise to infinity (curved).
X-H: heavy atom (x) is stationary and light atom (H) moves, vibrating as a simple harmonic oscillator. Bonds act like springs.
HO for diatomics
Atoms with similar masses move relative to each other.
Push atoms together (e- repel, increase E) and pull apart (until bond breaks) when vibrating with respect to r(eq).
HO: models bottom of potential well well, but not at higher E.
Parabolic V: close to r(eq) can approx V by parabola (V=1/2kx²), x=r-r(eq). k is a measure of curvature of V close to eqm extension of bond. Steep walls = stiff bond = large k.
Frequency of oscillator depends on the two masses
-homonuclear (F2): m1=m2
-heteronuclear (NaCl): m1 doesn’t equal m2
SWE: -ℏ²/2m(eff) . (1/2x²)Ψ = EΨ
E=(v+1/2)ℏω, v=0,1,2… and ω=root(k/m(eff))
Curvature of walls leads to some Ψ extending outside of the parabola (tunnelling). ΔE=ℏω, so E level spacings are even.
Tunnelling - probability of being in a forbidden region, independent of k and m.
ex: inversion of NH3. Doesn’t go over E barrier of planar form (not superimposable). Absorption ~900cm-1 (barrier ~2000cm-1), corresponding to v0->v1 (lowest vibrational state).
Effective mass (M(eff))
Measure of mass moved during vibration
m(eff)=m1m2/(m1+m2)
Reduced mass
Emerges from separation of relative internal and overall translational motion. Meff and u are not the same overall (except have same values for diatomics).
Vibrational term (G~)
Expresses E of vibrational states in terms of wavenumber (cm-1). E=hcG~(v)
G~(v)=(v+1/2)v~
v=ω/2π and v~=ω/2πc
IR spectroscopy
Frequency of E level transition (v~10E14 Hz). Use IR to look at vibrations of molecules.
Molecular fingerprint
-k proportional to bond strength
-bond strength depends on molecular environment
-selection rule: for a vibration to be observed, the dipole must change (heterogeneous, asymmetric or bent)
Anharmonic Oscillators (AO)
AOs can break bonds to make new bonds (limitation of HO - based on a parabola). HO describes the small changes in bond length well (parabola), but is wrong for increased E/Ψ.
At higher vibrational excitations, movement of atoms allows molecules to explore regions of V curve where parabolic approx. is poor. F no longer proportional to x.
Curve less confined than parabola. E levels less widely spaced at high excitations (weaker bond) - Morse potential
Morse Potential
V=hcD(e){1-e(-a(r-r(e)))}²
-a=root(m(eff)ω²/2hcD~(e)) - width of potential
-r - internuclear separation
-D~(o) - dissociation energy from v=0 to highest occupied v
-D~(e) - dissociation energy from r(eq) to highest occupied v
D(e)-D(o) = ZPE
Near minimum vibration of v with displacement resembles a parabola. Allows for dissociation at large displacements.
Permitted E levels: G~(v)=(v+1/2)v~-(v+1/2)²X(e)v~
-X(e) = a²ℏ/2m(eff)ω = anharmonicity constant
-G~(v) becomes smaller, reaching v(max) at G~=0.
Δv=+/-1, so ΔG~(v+1/2)=G~(v+1)-G~(v) -> converge
Dissociation limit - point where v=v(max). The separation between E levels vanishes and ΔG(v+1/2)=0.
v(max)=(v~/2v~X(e))-1
D(o) - when several vibrational transitions are detectable, can determine D(o) of bond. Sum of successive intervals from v=0 to v(max).
Birge-Sponer Plot - plot ΔG~(v+1/2) against v+1/2. X-axis intercept will give v(max). Inaccessible part of spectrum estimated by linear extrapolation. Integrate to get D(o) - often overestimated as realistically not linear.
HO vs MO
HO
+gives bond strength; apply to polynuclear systems
-doesn’t give D(o) and doesn’t fit experimental data well
MO
+gives bond strengths and D(o) and fits experimental data well
-cannot be applied to polyatomic molecules
Particle on a sphere
Ψ must satisfy 2 cyclic boundary conditions, resulting in 2 quantum number’s for state of angular momentum. θ and ϕ. Ψ must match over the poles (θ) and round the equator (ϕ)
SWE: -ℏ²/2m . ∇² = EΨ
-∇² (laplacian) = d²/dx²+d²/dy²+d²/dz²
-Once object is set in motion in a vacuum, no E required to keep it spinning, so V=0.
Rotational qn’s: 2 orbits -> 2 qn’s
L=angular momentum qn, L=0,1,2..
m(l) = component of orbital angular momentum about the x-axis, +L to -L.
E particle: restricted to certain values due to boundary conditions, continuous Ψ around sphere. Independent of m(l).
E=L(L+1)ℏ²/2I, where I=mr²
Classical model - centre of mass depends on m1/m2
-homogenic diatomic - m1=m2
-heterogenic diatomic - m1r1=m2r2 and r=r1+r2
Rigid Rotor
Fixed radius (rotation doesn’t result in stretching of the bond). Thus, a body does not distort under the stress of rotation, due to negligible centrifugal force.
Simplification, use μ to transform problem to that of a single mass rotating around the origin.
μ=m1m2/m1+m2 x 1.66o54E-27
I depends on masses of atoms present and molecular geometries. 3 I defined about 3 perpendicular axes: I(c)>I(b)>I(a)
Linear rotor - E quantised.
E(J)=J(J+1)ℏ²/2I, J=0,1,2…
E level ladder: E expressed in terms of a rotational constant (B~)
B~=ℏ/4πcI or hcB~=ℏ/2I
E(J)=hcB~J(J+1)=nBJ(J+1)
-E level spacings get further apart with increasing J.
-B expressed as frequency = B~c
E rotational state: rotational term F~(J)
F~(J)=B~J(J+1)=E(J)/hc
Frequency: E=hb=hv
Rotational-Vibrational Spectrum
Heteronuclear diatomics in the gas phase.
Rotational and vibrational motion is not discrete and combine to give special spectral information. As the molecule vibrates, the bond length changes. Rotational E changes with bond length (longer bond = slower spin).
Bond spectrum - high resolution vibrational spectrum has bands. n peaks = 2 isotopes (more abundant = bigger)
Selection rule: ΔJ=+/-1 and sometimes ΔJ=0 during a vibrational transition.
Spectral branches - each vibrational E level has a series of rotational E levels associated with it. Appearance of S~.
S~=G~(v)+F~(J) = (v+1/2)v~+B~J(J+1)
Vibrational transition from V+1
P Branch
ΔJ=-1
Transition: v+1
Q Branch
ΔJ=0
Transition: v+1
R Branch
ΔJ=+1
Transition: v+1
