Questions 8 - 12
Q8 bra-ket
Using the bra-ket notation, work out the following matrix multiplications in the orthonormal basis of an s and two p orbitals , for the linear combination of wavefunctions
Assume that and are normalized, which means for example that, and similarly for coefficients .
(a) Calculate and and similarly for the other basis vectors and show that the basis is orthonormal.
(b)
Strategy: Check that the matrices are commensurate for each calculation, the column of the left matrix (or vector) must have the same number or rows as the right matrix (or vector), and if so multiply them out. The next important point is to note that in vector form the standard basis sets are orthogonal and normalized, which means they are
and the wavefunctions the kets;
Q9 bra-Ket
Using and defined in question 8, show that is a general statement for any bra-ket pair.
Q10 Proton NMR
In the NMR spectroscopy of a single H, with spin quantum number, a two element basis set can be defined as to represent the spin state. The spin state is represented as the column vector and as this is a basis element (basis ket) of the standard basis set it is zero except for one element which is unity. Similarly is represented as the column vector .
The angular momentum operator has components in the -directions and because it is an operator it is represented as a matrix. Angular momentum has units of and the components of the angular momentum operators can be represented as the (Pauli) matrices,
(a) Show that if then
where is the unit diagonal matrix.
(b) Show that and commute, , but that and the other combinations of components do not commute.
(c) Show that when operates on , the eigenvalue is , i.e. show that and calculate and . Comment on the results. These equations are eigenvalue-eigenvector equations, as is the Schroedinger equation where is the wavefunction, the energy, and the operator.
(d) Calculate and similar terms for the and components.
(e) Using the operators and , show that these are raising or lowering operators which convert the eigenstate or into the other.
Q11 Raising - lowering operators
Angular momentum raising and lowering (shift) operators move a system from one state to another. They have the property
where the angular momentum quantum number is and the projection or quantum number is . The values takes run from in unit steps.
(a) Show that the basis set used is orthonormal.
(b) Calculate the raising and lowering operators for a state with angular momentum in the basis set. Note the representation of the operators will be a matrix.
(c) Using the result from (a) show that the commutator where is given by equation 21. (assume and change ).
(d) Calculate the operator for spins, and then for half unit spins in the basis set.
Strategy: (b) The basis set must contain all values therefore the set could be
if . Define vectors so that they are orthonormal. (c) The raising and lowering operators form a block diagonal matrix. Individual blocks of which can be calculated as in (a) or a more extensive basis set formed and the whole matrix calculated. It may be useful to define a python/Sympy function to work out terms in the operator according to the equations.
Q12 Raising - lowering operators
(a) Using the raising and lowering operator in the previous question produce equations 19, 20 starting from
(Assume .)
(b) Show that the commutator is
where is given by a similar equation to 21,