Quantum Mechanics Flashcards
What is the wavefunction?
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
Square modules of the wavefunction is proportional to …..
The probability denisity. (gives likelihood of where particle is most likely to be found)
Schrodinger is the
total energy of the system, kinetic energy + potential energy.
Eigenfunction of the Schrodinger are
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.
(operator A)(operator B)(f(x)) =
(operator B*f(x)) and then operatre on it with A. Do not commute!
Two operators are equal if
They have the same effect on any operand
Commutator of 2 operators A and B [A, B] =
AB - BA. If [A,B] = 0 the operators commute. [B,A] = -[A,B]
An operator A is linear if….
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)
Normalisation
integral(psipsi) = integral(mod of psi^2) = I = 1. Ensures total probability of particles being somewhere = 1. Choose N = I^(-1/2) and Npsi.
Potential energy operator is
Interaction between charges and total potential energy for all the particles in the system.
Harmonic oscillator has
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.
Reducing the motion for vibrating diatomic:
Translational mass of whole molecule M = m1 +m2 or vibrational motion of nuclei as a single particle with reduced mass.
KE operator for vibration =
-hbar/2(reduced mass) *d2/dx^2. x = distance between two atoms.
Reduced mass
(m1m2) / (m1+m2)
Raising operator
Let us generate a new eigenfunction of H with a larger eigenvalue
[A(operator), c] =
0 any linear operator commutes with a constant
Q dagger is
raising operator for the harmonic oscillator. If have lowest eigenfunction then we can generate all the rest.
Q is
lowering operator of harmonic oscillator
lowering operator acting on wavefunction =
eigenvalue of lowest energy - Q*wavefunction = 0
ground state energy of simple harmonic oscillator
1/2
energy level expression for simple harmonic os cillator
v + 1/2
levels of harmonic oscillator are
equally spaced with zero-point energy equal to half the spacing
nth wave function =
weighted sum over all basis sets in wavefunction of coefficient of given contribution of each basis function to overall function times basis function
How to find set of coefficients that give the best representation of nth wavefunction
construct a matrix and diagonalise
kronecker delta ij =
1 if i = j and 0 if i does not = j
secular equation
each eigenfunction of H operator is specified by column vector with components c(jn) which define a wavefunction through basis set expansion
if N basis function in basis set there are
Nsolutions and N eigenvalues
if basis set form orthonorma; basis the overlap function =
kronecker delta function
Solving Schrodinger by matrix
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
eigenvector matrix C is made up of
N eigenvectors c(n), each eigenvector is a column vector with N components
nth wavefunction represented by
weighted sum of basis functions. orthogonal to one another and may by chosen to be normalised.
if the basis functions are orthonormal then
different eigenvectors are orthogonal and each eigenvector is normalised
Functions phi_j (the basis set) are functions we know
before we start
To give best possible represention of wavefunction_n we find a
set of coefficients c_jn by constructing a matrix and diagonalizing it
kronecker delta = 1 if
i = j and 0 if i is not j
if N basis functions in the basis set there will be
N solutions (c_n) and N eigenvalues
HC =
SCE
To solve Schrodinger equation by constructing and diagonalising matrics you :
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
Eigenvector matric C is made up of N
eigenvectors c_n where each eigenvector is a column vector with N compnonents c_jn
If basis sets are orthonormal then eigenvalues are orthonormal and overlap integral between 2 coeffs is equal to
kronecker deta
if C is real it is
orthogonal matrix
Hermitian conjugate of C times C equals
identity (unitary matrix)
if coefficient is higher for one atom the bonding orbital contains
more of higher atom character, polarised towards that atom. orbital contributes more than twice as much electron density on that atom
the ket is a
state vector na dimplied it is normalised
no wavefunctions inside
bra-kets
Hermitian operator means bra-ket =
conjugate of reordered expectation value
state vectors are
orthonormal
Hermitain operator properties
eigenvalues are real and eigenfunctions corresponding to different eigenvalues are orthogonal
expectation value of hamiltonian calculated using an approximate wavefunction is always
greater than or equal to the true ground state energy of the system
If two expectation values is compared the lower one is always
closer to real value
Uncertainty principle says
we can know the values of 2 observables simultaneously and precisely only if they commute
Can only measure precisely if the wavefunction is
eigenfunction of operator (every measurement gives same answer)
if a function is a nondegenerate eigenfunction of A and A and B commute then function is also
an eigenfunction of
if a function is a nondegenerate eigenfunction of A and A and B commute then function is also
an eigenfunction of B
if operators do not commute the product of uncertainties (product of standard deviations of distribution about their mean value) is limited by
1/2mod(<[A,B]>)
when one particle rotates about another the angular momentum is
avector along the axis of rotation
Angluar momenta add together
vectorially
J (spin-orbit coupling) =
L +S, J takes values |L-S| to |L +S| in steps of 1
3 components of an angular mometnum cannot be known
simultaneously
anglar momentum vector precessing about
z axis
if 2 particles are indistinguishable the system must appear the same if they are
exchanged (density must be the same)
symmetric function is unchanged when particles are
exchanged
when particles are exchanged the antisymmetric function
changes sign
a single electron always has spin -
1/2 with two stae (spin up/spin down m_s = +-1/2)
PAULI PRINCIPLE: total wavefunction of a 2 electron system is a products of its
spatial and spin parts
total wavefunction for system is always antisymmetric with respect to exchange of
identical fermions
total wavefunction for system is always symmetric with respect to exchange of
identical bosons
electrons and particles with half integer spins are
fermion
bosons are
photons and particles with integer spin (even mass nuclei)
singlet (antisymmetric)spin functions must be paired with
symmetric space functions
triplet (symmetric) spin functions must be paired with
antisymmetric space functions
spatial part of the wavefunction is different for singlet and triplet states arising from the same
electron configuration(one electron in orbital a and other in orbital b)
Pauli exclusion principle:
it is not possible to place 2 electrons with parallel spins in the same orbital
Hunds rule
when 2 atomic or molecular states arise from the same electron configuration the one with higher spin multiplicity is lower in energy
Fermi hole: 2 electrons with parallel spin
avoid each other
due to coloumb repulsion (arising from symmetry associated with spin states)between electrons keeping them apart
lowers the energy
2 electrons with spins paired (anti singlet, sym spatial) tend to
pile together
What is an operator?
Mathematical entity which acts on function
