Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Core 4b > Quantum Mechanics > Flashcards

Quantum Mechanics Flashcards

1
Q

What is a harmonic wave?

A

A wave with a frequency that is a whole number

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

What is a black body?

A

The hypothetical “perfect” absorber and emitter of EM radiation (across all frequencies)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

How did Max Planck’s “quanta” contradict classical physics?

A

His quanta were discrete: they had fixed energy values instead of continuous values like classical physics

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

What are the two components of the Schrodinger equation for a particle in 1 dimension?

A

The kinetic energy and the potential energy

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

How does the Schrodinger equation differ for 3 dimensions than for 1 dimension?

A

The 3D equation has partial derivatives for each coordinate axis (x,y,z)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is the Born interpretation of wavefunction?

A

The probability of finding the particle in the interval Δx (or dx) is proportional to the wavefunction squared (probability density) multiplied by the interval: ψ^2 (x) * Δx (IN 1D)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Why should a wavefunction be normalised?

A

So that the probability of finding a particle anywhere in space is equal to 1 (100% likely)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What are the 5 criteria for an acceptable wavefunction?

A

The wavefunction must:
1) have only one value at any point
2) not be infinite over a measurable interval
3) be a continuous and smooth function
4) have as first derivative that is a continuous and smooth function
5) the integral ψ^2 between infinity and negative infinity must be finite

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Mathematically, what is the uncertainty principle?

A

ΔpΔx >= 1/2ħ
therefore a particle must always have error in its momentum and its position

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

What are the steps in calculating the allowed energies of a particle inside a 1D box?

A
  • Define the potential energy component (V=0)
  • Solve the Schrodinger equation (Asin(kx) + Bocs(kx))
  • Apply boundary conditions (0% probability at x=0 and x=L)
  • Solve for k and A
  • Normalise so the probability is 100% inside the box
  • Determine allowed energies using E = ħ^2 k^2 / 2m
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

What are orthogonal wavefunctions?

A

The overlap integral of two different quantum numbers = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

What is the bra-ket notation for the integral of xy?

A

{x|y}

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What is the free electron model for the electronic structure of polyenes?

A

H2C=CH-CH=CH2. Pi electrons move along the chain like particles in a 1D box. The potential energy along the chain is constant but rises sharply at the end

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

How would u calculate the energy of the ground state of butadiene? (C4H6)

A

E = (n^2 h^2) / (8m L^2)
There are 4pi electrons; 2e for n=1 and 2e for n=2
Total E = 2E (where n=1) + 2E ( where n=2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

How does the barrier in quantum tunnelling differ in comparison to the barrier in a 1D box?

A

The barrier is not of infinite potential and therefore has a probability of tunnelling through (it is a finite barrier)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

What happens to the amplitude and frequency of a wavefunction after it tunnels through a finite barrier?

A

The frequency remains the same but the amplitude is lowered

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

How does the Schrödinger equation for a particle in a 3D box differ to the equation for a 1D box?

A

It has partial derivatives for the 3 coordinate axis

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

What are degenerate energy levels?

A

Energy levels with the identical energies but different wavefunctions: (2,1,1), (1,2,1) and (1,1,2) all have identical energies but different wavefunctions. (nx, ny, nz)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Give an example of a non-degenerate energy level

A

(1,1,1), (2,2,2), (3,3,3), ….

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

What is the equation for calculating an arc length?

A

s = r θ
where:
r = radius
θ = angle

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

How does the Schrödinger equation for a particle on a ring differ to the equation for a 1D box?

A

x^2 becomes (rθ)^2
I = m r^2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

What is probability density independent of for a particle in a ring?

A

The angle θ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

What is the equation for classical angular momentum and quantum angular momentum?

A

L = m*r^2
L = pr

L = m⌄L * ħ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Why is calculating the probability density for a particle on a ring different to a 1D box and how is it calculated?

A

The wavefunction for a particle on a ring is complex and therefore its complex conjugate must be used to calculate the probability density. For the complex conjugate change the sign before the imaginary value

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

How does the Schrödinger equation for the quantum harmonic oscillator differ to that of a 1D box?

A

It includes the potential energy in the spring:
V = 1/2 k x^2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

What can the oscillation of a reduced mass be compared to?

A

The harmonic oscillation of two masses (like a diatomic molecule)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

In the Schrödinger equation what is the name of the operator, the eigenvalue and the eigenfunction of the operator?

A

Operator: Hamiltonian operator
Eigenvalue: energy
Eigenfunction: wavefunction

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

What does the Hamiltonian operator include?

A

The kinetic and potential energy

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

What is a Hermitian operator?

A

An operator that has a real eigenvalue (observable eigenvalues are real)

30
Q

For hydrogenic atoms, what is the potential energy?

A

The Columbic interaction between a positive mass (nucleus) and a negative mass (electron)

31
Q

How does the Shrodinger equation change, for a hydrogenic atom, for when we assume that the nucleus DOES in fact move?

A

We use a reduced mass instead of the mass of an electron (however because the proton is 1836x bigger than an electron it doesn’t make a huge difference)

32
Q

What coordinates are used when solving the Schrödinger equation for a hydrogenic atom?

A

Spherical polar coordinates:
r: radial distance
θ: zenith angle
ϕ: azimuth

33
Q

What are the 2 components when solving the Schrödinger equation for a hydrogenic atom?

A

1) Radial Wavefunction
2) Spherical harmonics

34
Q

What quantum numbers does the radial wavefunction component depend on?

A
  • the principle quantum number n
  • orbital angular momentum quantum number L
35
Q

What quantum numbers does the spherical harmonics component depend on?

A
  • orbital angular momentum number L
  • magnetic quantum number mL
36
Q

What is an atomic orbital (AO)?

A

A one-electron wavefunction for an atom

37
Q

What is significant about atomic orbitals with the same principle quantum number?

A

They are in the same electron shell

38
Q

What is significant about atomic orbitals with the same principle quantum number but different angular momentum quantum numbers?

A

They are in the same electron shell but in different sub-shells e.g. 4s and 4p

39
Q

The s-orbital depends only on the distance from the nucleus. What is significant about that?

A

It gives rise to the spherical shape of the s-orbital

40
Q

Which p-orbitals are real and which are imaginary?

A

px - imaginary
py - imaginary
pz - real

41
Q

What is a nodal point?

A

A change in sign of a wavefunction/isosurface plot

42
Q

Which d-orbitals are real and which are imaginary?

A

dz^2 is real and the rest are imaginary

43
Q

Which two quantum numbers describe electron spin?

A
  • Spin quantum number = 1/2
  • Spin magnetic quantum number = +1/2 OR -1/2
44
Q

For an alpha and beta electron, give both their z-axis direction and spin magnetic quantum number

A

alpha: +1/2 ↑
beta: -1/2 ↓

45
Q

For alpha and beta spin functions, what does a value of ω = +1/2

A

α(+1/2) = 1
β(+1/2) = 0
For -1/2 the opposite is true

46
Q

What are the components involved in calculating the energy of two particles, a and b?

A

E = T(a) + T(b) + V(a) + V(b) + V(a,b)
Need to take into account the potential energy interaction between the two particles

47
Q

What is the orbital approximation?

A

A method of visualising electron orbitals for a chemical species with more than 3 quantum particles. Assumes every electron has its own spin orbital

48
Q

What is the Pauli Exclusion principle?

A

“No more than two electrons may occupy a given spatial orbital, and if two electrons do occupy the same spatial orbital, then their spins must be opposite”

49
Q

What does the variation principle tell us?

A

The energy of an approximate wavefunction is always higher than that of the exact solution to the Schrodinger equation

50
Q

What is a Slater determinant?

A

A determinant of spin orbitals that changes sign when changing the position of the electrons - therefore it fulfils the Pauli exclusion principle for an approximate wavefunction

51
Q

What is electron shielding?

A

An electron at a distance form a nucleus in a many-electron atom is shielded and experiences a reduced nuclear charge (Z eff)

52
Q

What is penetration?

A

Looking at a radial distribution function; s-orbitals penetrate closer to the nucleus and experience less shielding than a p-orbital and therefore s-orbitals have a lower energy than p-orbitals in the same shell

53
Q

What is the aufbau principle?

A

Electrons fill up the lowest available energy sub-shells first before moving to higher energy sub-shells

54
Q

What is Hund’s rule?

A

Electrons occupy different orbitals of a given sub-shell before doubly occupying any one of them

55
Q

What is the Born-Oppenheimer approximation?

A

Electrons and nuclei move independently.
Fast-moving electrons move around stationary nuclei.
Slow-moving nuclei experience the averaged influence of the fast-moving electrons.

56
Q

What does the Born-Oppenheimer approximation allow us to do to the Schrodinger equation?

A

It allows separation of the Schrodinger equation into two equations; one for electrons and one for nuclei

57
Q

What graph can we obtain when using the Born-Oppenheimer approximation for a diatomic molecule and what does the graph show?

A

A potential energy curve - it shows repulsion at small distances between nuclei, the equilibrium bond length and the lack of attraction at large distances

58
Q

What graph can we obtain when using the Born-Oppenheimer approximation for multi-electron molecule and what does the graph show?

A

A potential energy surface - local minima, first- and second-order saddle points, a valley ridge inflection point and reaction paths

59
Q

In regard to the Schrodinger equation for an H2 molecules, why do we not include the kinetic energy of the nuclei?

A

We use the Born-Oppenheimer approximation which says that the nuclei are fixed in space

60
Q

What are the components of the Schrodinger equation for an H2 molecule?

A

H(1,2) = T(1) + T(2) + V(1) + V(2) + V(1,2) + V(A,B)
where T is kinetic energy, V is potential, 1 and 2 are electrons and A and B are nuclei

61
Q

What is LCAO?

A

Linear combination of atomic orbitals - an approximation of a molecular orbital
ψ(MO) = c(a)ψ(a) + c(b)ψ(b)
where ψ(a) and ψ(b) are 1s-orbitals
and c(a) and c(b) are abstract coefficients

62
Q

During LCAO, it is calculated that c(a) = ± c(b). What is the implication of this?

A

Two atomic orbitals can be formed - one of lower energy and one of higher. The lower energy orbital is a bonding orbital and the higher energy one is an anti-bonding orbital

63
Q

Is the anti-bonding orbital in H2 gerade or ungerade?

A

Upon inversion, through the inversion centre, the sign of the orbital WILL change and therefore the anti-bonding orbital is ungerade. The opposite is true for the bonding orbital

64
Q

Is the difference in energy of the bonding orbital, compared to the atomic orbital, smaller or larger than in the anti-bonding orbital?

A

The bonding orbital has a smaller decrease in energy than the anti-bonding orbital has an increase in energy

65
Q

Why is helium a monoatomic gas?

A

A diatomic helium molecule would have to place electrons in the anti-bonding orbital and because the energy gap for the anti-bonding orbitals is greater than the bonding orbits, so repulsion occurs

66
Q

In regard to period 2 elements, 1s AO are very small and compact with lower energies than the 2nd shell electrons. What can we assume about electrons in the 1s orbitals?

A

They form a pair or bonding and anti-bonding orbitals and do not interact with the 2nd shell orbitals

67
Q

For period 2 elements, what are all the possible combinations for orbital overlap for a diatomic molecule?

A

2s-2s, 2s-2pz, 2pz-2pz, 2pz-2s. Where the first orbital is atom A and the second one is atom B.

68
Q

For a diatomic period 2 element molecule, how many MOs are there?

A

4σ orbitals and 4π orbitals

69
Q

What is s-p mixing?

A

The bonding pz orbital decreases in energy along the period until O2 where it drops below the degenerate bonding pi orbitals

70
Q

What is paramagnetism and diamagnetism?

A

Molecules with two electrons with unpaired parallel spin (↑↑) are paramagnetic and are drawn into a magnetic field. Diamagnetic compounds have paired spin electrons (↑↓)

71
Q

What is the equation for calculating bond order?

A

b = 1/2(n-n)
where n = number of electrons in bonding orbitals
n = number of electrons in anti-bonding orbitals

Core 4b (4 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions