Quantum mechanics Flashcards
(45 cards)
What is the equation for kinetic energy?
K = 1/2 mv^2 = p^2/2m
where p = (h bar) x a
Equation for average kinetic energy of a monoatomic gas i.e. argon
1/2 kB T
where kB = Boltzmann constant
T = absolute temperature
kBT = thermal energy
What is potential energy?
Energy required to place particles of system in given position i.e. energy associated to position of such particles
Give examples of potential energy and any laws used to give potential energy (give equations)
Example: Potential energy between two charges q1 and q2 at distance r given by Coulomb Law
Vc(r) = e^2/(4piε0) x q1q2/r
where e = elementary charge
ε0 = 8.854 x10^-12 F.m-1
Example 2: Potential energy between 2 atoms that can form chemical bond
Compute thermal energy at T = 300K and express in J and eV
kBT = 1.381 x10^-23 JK-1 x 300K = 4.143 x10^-21 J –> 0.02586 eV = 25.85 meV
J/1.602 x10^-19 = eV
Compute linear momentum of argon atom (m=40 u) or an electron (m = 9.1 x10^-31 kg) if their kinetic energy is equal to thermal energy computed in previous question (4.413 x10^-21 J)
p = mv
m = 40 x 1.661 x10^-27 kg = 6.644 x10^-26 kg
K = p^2/2m
For argon atom:
p = sqrt(K x 2m) = sqrt(4.413x10^-21 x 2(6.644x10^-26)) = 2.422 x10^-23 kg m/s
For electron:
p = sqrt(4.413x10^-21 x 2(9.1x10^-31)) = 8.962 x10^-26 kg m/s
Description of a classical system of N particles (Law involved)
- Position and velocity at initial time
- Forces or potential energy between particles
- Mass of all particles (m1, m2, …)
- Newton’s Law (Fi = miai)
- System fully descirbed by {r1(t), r2(t), …, rN(t), v1(t), v2(t), …, vN(t)}
where rN(t) = position of all particles as a function of time
vN(t) = velocity of all particles as a function of time
Description of a quantum system of N particles (equation involved, requirements of wavefunction)
- Wavefunction at an initial time
- Potential energy of a system
- Mass of all particles (m1, m2, …)
- Schroedinger equation
- System fully described by wavefunction ψ(r,t)
where r = coordinates of all particles and t = time - Wavefunction must be continuous (no jumps) and differentiable (no corners)
- Values of wavefunction can be complex (with real and imaginary part)
Where are the particles at a given time in a quantum system? (probability of finding)
- System with 1 particle in 1 dimension (1 coordinate x) has wavefunction: ψ(x,t)
- Probability of finding particle in vicinity of certain value of x is: |ψ(x,t)|^2
- Quantum mechanics only predicts probability of finding particle in given position
Compare/contrast classical mechanics with quantum mechanics
Classical:
- Time evolution: trajectory (i.e. position and velocity as function of time)
- Info needed: Initial position/velocity at t = 0, potential energy V, mass of particles
- Position of particles: Known with certainty
- Other observables: Known with certainty
Quantum:
- Time evolution:wavefunction (i.e. function of all coordinates and time)
- Info needed: Initial wavefunction at t = 0, potential energy V, mass of particles
- Position of particles: Known as a probability
- Other observables: Known as a probability (in general)
What is an operator?
- A mathematical object (O hat) that transforms a function into another function
- In O hat f = g, O hat transforms function f into new function g
- Operator that transforms function into derivative written as O hat 1 = d/dx
- Operator that transforms function into same function multiplied by x written as O hat 2 = x
- Operators can be summed or multiplied
What is the de Broglie relation?
lambda = 2 pi h bar/p = h/p
Describe the particle in a box theory (give equation for free particle wavefunction)
- Particle of mass m in 1 dimension can only occupy positions between x = 0 and x = L
- Wavefunction must = 0 for x<0 and x>L (particle is never there)
- Wavefunction will also be 0 at edge of box as it must be continuous
- In region 0<x<L particle is free and free particle wavefunction can be used ψ(x) = Acos(ax) + Bsin(ax)
What is the normalisation of a wavefunction?
Multiplying the wavefunctions by a factor such that the integral over all space is 1
What are observables? Give examples. When are they used/cannot be used?
Everything you can actually measure (classical):
- Position (x)
- Linear momentum (p(x))
- Kinetic energy (K)
- Angular momentum (L)
- Wavefunction cannot be measured (quantum mechanics).
How are observables and operators related and give examples.
- Correspondence between everything you can measure and a mathematical operator
- Operators can be seen as variables that multiply by function to give different function
Operators:
- Position x: x hat –> x
- Linear momentum p(x): p(x) hat –> -ih bar d/dx
- Kinetic energy: K hat = -(h bar)^2/2m x d^2/dx^2
What is expectation value? Give equation
- If system in generic state psi (not eigenfunction of A hat) and one measures quantity of A, measure gives one of eigenvalues of A with certain probability
- Expectation value = average of many measures
- <a> = (integral of psi * A hat psi d tau)/(integral of psi * psi d tau)</a>
What is the Heisenberg Uncertainty Principle?
- Different operators/observables have different eigenfunctions
- It is not possible to measure both quantities with arbitrary accuracy
- Uncertainty set by quantum mechanics
- For position and momentum: delta(x)delta(px) is greater than or equal to 1/2h bar
- For smaller boxes (decreasing L), become more sure of position (delta x decreases) and more unsure about momentum (delta p increases)
What is the lowest energy level in the harmonic oscillator and why (give equation)?
- 1/2 h bar omega
- Can’t be 0 (bottom of potential energy curve) because equation Ev = h bar omega (1/2 + v)
- Would mean position and momentum of particle known with arbitrary precision which is forbidden by Heisenberg principle
Describe harmonic oscillator energy graph - how are the energy levels spaced and what is the space between them?
- Evenly distributed
- h bar omega between each energy level Ev (v = 0,1,2, etc)
- v starts from 0, whereas in particle in box they start from 1
Equation for wavelength
lamda = c/v
c = speed of light
v = frequency
Equation for angular frequency
omega = 2.pi.v
How is radiation absorption measured? (3 equations)
Transmittance: T = I/I0 (transmitted light/incident light)
Absorbance: A = -logT
Beer-Lambert Law: Av = 𝜀v.cL
What is rate of light absorption proportional to? Which transitions are allowed/forbidden?
proportional to |𝜇(ij)|^2
transition dipole moment 𝜇(ij) = integral of 𝜓(𝑖)∗𝜇𝜓(𝑗) 𝑑𝜏
where 𝜇 = dipole moment operator
𝜇(ij) = 0, transitions FORBIDDEN
𝜇(ij) ≠ 0, transitions ALLOWED
