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Computation Chemistry > Quantum Mechanics > Flashcards

Quantum Mechanics Flashcards

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1
Q

What is the wavefunction?

A

It describes the state of a system. It tells us where the particle is localised in space. Tells us probability of the particle being at that point in space

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2
Q

Square modules of the wavefunction is proportional to …..

A

The probability denisity. (gives likelihood of where particle is most likely to be found)

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3
Q

Schrodinger is the

A

total energy of the system, kinetic energy + potential energy.

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4
Q

Eigenfunction of the Schrodinger are

A

a constant (allowed energies of the system) x wavefunction. Constant (E) are the eigenvalues. Eigenfunctions that differ only by a costant factor describe the same state.

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5
Q

(operator A)(operator B)(f(x)) =

A

(operator B*f(x)) and then operatre on it with A. Do not commute!

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6
Q

Two operators are equal if

A

They have the same effect on any operand

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7
Q

Commutator of 2 operators A and B [A, B] =

A

AB - BA. If [A,B] = 0 the operators commute. [B,A] = -[A,B]

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8
Q

An operator A is linear if….

A

A(c1f1 +c2f2) = c1Af1 +c2Af2 (c is complex number, f is function). Acf = cAf (special case) - if f is an eigenfunction of A than a constant times f is an eigenfunction with eigenvalue a: A(cf) = a(cf)

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9
Q

Normalisation

A

integral(psipsi) = integral(mod of psi^2) = I = 1. Ensures total probability of particles being somewhere = 1. Choose N = I^(-1/2) and Npsi.

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10
Q

Potential energy operator is

A

Interaction between charges and total potential energy for all the particles in the system.

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11
Q

Harmonic oscillator has

A

equally spaced energy levels up to disssociation. Zero-point energy equal to half the spacing between levels. Wavefunctions for harmonic oscillator tunnel into regions where V(x) > E. Not zero probability of finding particle outside well.These regions are classically forbidden because KE cannot be negative.

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12
Q

Reducing the motion for vibrating diatomic:

A

Translational mass of whole molecule M = m1 +m2 or vibrational motion of nuclei as a single particle with reduced mass.

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13
Q

KE operator for vibration =

A

-hbar/2(reduced mass) *d2/dx^2. x = distance between two atoms.

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14
Q

Reduced mass

A

(m1m2) / (m1+m2)

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15
Q

Raising operator

A

Let us generate a new eigenfunction of H with a larger eigenvalue

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16
Q

[A(operator), c] =

A

0 any linear operator commutes with a constant

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17
Q

Q dagger is

A

raising operator for the harmonic oscillator. If have lowest eigenfunction then we can generate all the rest.

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18
Q

Q is

A

lowering operator of harmonic oscillator

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19
Q

lowering operator acting on wavefunction =

A

eigenvalue of lowest energy - Q*wavefunction = 0

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20
Q

ground state energy of simple harmonic oscillator

A

1/2

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21
Q

energy level expression for simple harmonic os cillator

A

v + 1/2

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22
Q

levels of harmonic oscillator are

A

equally spaced with zero-point energy equal to half the spacing

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23
Q

nth wave function =

A

weighted sum over all basis sets in wavefunction of coefficient of given contribution of each basis function to overall function times basis function

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24
Q

How to find set of coefficients that give the best representation of nth wavefunction

A

construct a matrix and diagonalise

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25
Q

kronecker delta ij =

A

1 if i = j and 0 if i does not = j

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26
Q

secular equation

A

each eigenfunction of H operator is specified by column vector with components c(jn) which define a wavefunction through basis set expansion

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27
Q

if N basis function in basis set there are

A

Nsolutions and N eigenvalues

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28
Q

if basis set form orthonorma; basis the overlap function =

A

kronecker delta function

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29
Q

Solving Schrodinger by matrix

A

choose basis set containing N basis function. evaluate matrix elements H(ij) and S(ij) and construct N X N matrices H and S. Solve HC = SCE or HC = CE depending if orthonormal to find eigenvalues and eigenvectors

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30
Q

eigenvector matrix C is made up of

A

N eigenvectors c(n), each eigenvector is a column vector with N components

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31
Q

nth wavefunction represented by

A

weighted sum of basis functions. orthogonal to one another and may by chosen to be normalised.

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32
Q

if the basis functions are orthonormal then

A

different eigenvectors are orthogonal and each eigenvector is normalised

33
Q

Functions phi_j (the basis set) are functions we know

A

before we start

34
Q

To give best possible represention of wavefunction_n we find a

A

set of coefficients c_jn by constructing a matrix and diagonalizing it

35
Q

kronecker delta = 1 if

A

i = j and 0 if i is not j

36
Q

if N basis functions in the basis set there will be

A

N solutions (c_n) and N eigenvalues

37
Q

HC =

A

SCE

38
Q

To solve Schrodinger equation by constructing and diagonalising matrics you :

A

Choose basis set with N basis functions. Evaluate matrix elements H_ij and S_ij. Solve HC = SCE or HC = CE and find eigenvalues/vectors

39
Q

Eigenvector matric C is made up of N

A

eigenvectors c_n where each eigenvector is a column vector with N compnonents c_jn

40
Q

If basis sets are orthonormal then eigenvalues are orthonormal and overlap integral between 2 coeffs is equal to

A

kronecker deta

41
Q

if C is real it is

A

orthogonal matrix

42
Q

Hermitian conjugate of C times C equals

A

identity (unitary matrix)

43
Q

if coefficient is higher for one atom the bonding orbital contains

A

more of higher atom character, polarised towards that atom. orbital contributes more than twice as much electron density on that atom

44
Q

the ket is a

A

state vector na dimplied it is normalised

45
Q

no wavefunctions inside

A

bra-kets

46
Q

Hermitian operator means bra-ket =

A

conjugate of reordered expectation value

47
Q

state vectors are

A

orthonormal

48
Q

Hermitain operator properties

A

eigenvalues are real and eigenfunctions corresponding to different eigenvalues are orthogonal

49
Q

expectation value of hamiltonian calculated using an approximate wavefunction is always

A

greater than or equal to the true ground state energy of the system

50
Q

If two expectation values is compared the lower one is always

A

closer to real value

51
Q

Uncertainty principle says

A

we can know the values of 2 observables simultaneously and precisely only if they commute

52
Q

Can only measure precisely if the wavefunction is

A

eigenfunction of operator (every measurement gives same answer)

53
Q

if a function is a nondegenerate eigenfunction of A and A and B commute then function is also

A

an eigenfunction of

53
Q

if a function is a nondegenerate eigenfunction of A and A and B commute then function is also

A

an eigenfunction of B

54
Q

if operators do not commute the product of uncertainties (product of standard deviations of distribution about their mean value) is limited by

A

1/2mod(<[A,B]>)

55
Q

when one particle rotates about another the angular momentum is

A

avector along the axis of rotation

56
Q

Angluar momenta add together

A

vectorially

57
Q

J (spin-orbit coupling) =

A

L +S, J takes values |L-S| to |L +S| in steps of 1

58
Q

3 components of an angular mometnum cannot be known

A

simultaneously

59
Q

anglar momentum vector precessing about

A

z axis

60
Q

if 2 particles are indistinguishable the system must appear the same if they are

A

exchanged (density must be the same)

61
Q

symmetric function is unchanged when particles are

A

exchanged

62
Q

when particles are exchanged the antisymmetric function

A

changes sign

63
Q

a single electron always has spin -

A

1/2 with two stae (spin up/spin down m_s = +-1/2)

64
Q

PAULI PRINCIPLE: total wavefunction of a 2 electron system is a products of its

A

spatial and spin parts

65
Q

total wavefunction for system is always antisymmetric with respect to exchange of

A

identical fermions

66
Q

total wavefunction for system is always symmetric with respect to exchange of

A

identical bosons

67
Q

electrons and particles with half integer spins are

A

fermion

68
Q

bosons are

A

photons and particles with integer spin (even mass nuclei)

69
Q

singlet (antisymmetric)spin functions must be paired with

A

symmetric space functions

70
Q

triplet (symmetric) spin functions must be paired with

A

antisymmetric space functions

71
Q

spatial part of the wavefunction is different for singlet and triplet states arising from the same

A

electron configuration(one electron in orbital a and other in orbital b)

72
Q

Pauli exclusion principle:

A

it is not possible to place 2 electrons with parallel spins in the same orbital

73
Q

Hunds rule

A

when 2 atomic or molecular states arise from the same electron configuration the one with higher spin multiplicity is lower in energy

74
Q

Fermi hole: 2 electrons with parallel spin

A

avoid each other

75
Q

due to coloumb repulsion (arising from symmetry associated with spin states)between electrons keeping them apart

A

lowers the energy

76
Q

2 electrons with spins paired (anti singlet, sym spatial) tend to

A

pile together

77
Q

What is an operator?

A

Mathematical entity which acts on function

Computation Chemistry (3 decks)

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