Quantum Mechanics Flashcards by Anisha Badal

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

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Square modules of the wavefunction is proportional to …..

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

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Schrodinger is the

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

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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.

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

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

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Two operators are equal if

They have the same effect on any operand

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

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

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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)

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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.

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Potential energy operator is

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

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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.

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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.

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KE operator for vibration =

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

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Reduced mass

(m1m2) / (m1+m2)

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Raising operator

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

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[A(operator), c] =

0 any linear operator commutes with a constant

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Q dagger is

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

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Q is

lowering operator of harmonic oscillator

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lowering operator acting on wavefunction =

eigenvalue of lowest energy - Q*wavefunction = 0

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ground state energy of simple harmonic oscillator

1/2

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energy level expression for simple harmonic os cillator

v + 1/2

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levels of harmonic oscillator are

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

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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

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

construct a matrix and diagonalise

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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