eUsually helpful: when do not know where to start, review the definitions foritems in the problem.
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1.2 Are the linear combinations 2x - y - z, 2y - x - z, 2z - x - y linearly independent? .
I did not discuss it in class but it is easy to verify by recalling the definition of linear dependency: a set of vectors, v1, v2, v3, is linearly dependent if and only if there exists a set of scalars, a1, a2, a3, not all zero, such that a1v1 + a2v2 + a3v3 = 0. Matrix algebra give a simple recipe: construct a matrix out these column vectors (with numbers in front of x, y and z representing the coordinates) and evaluate its determinant; if it is zero - the vectors are dependent, otherwise - independent.
1.4 Show that (for the operators) (a) [A, B] = -[B, A], (b) [Am, An] = 0 for all m, n, (c) [A2, B] = A[A,B] + [A,B]A, (d) [A,[B,C]] + [B,[C, A]] + [C,[A,B]] = 0
Use the definition of a commutator and evaluate.
1.9 Find the operator for position x if the operator for momentum p is taken to be p = (h/4pm)1/2(A + B), with [A,B] = 1 and all other commutators zero. Hint. Write x = aA + bB and find one set of solutions for a and b.
You need two equations to identify unique a and b.One is the commutator [p,x] = -ih/2p (independent of represenattion). Plug your x in and get one relation between and b. Is there another eqaution?
1.11 Evaluate the commutators (a) [H, px] and (b) [H, x], where H = px2/2m + V(x). Choose (i) V(x) = V, a constant, (ii) V(x) = kfx2/2; (iii) V(x) --> V(r) = e2/4pe0r. Hint. For part (a, not b, as in the book), case (iii), use (dr-1/dx) = -x/r3
Evaluate each (a) and (b) in a general case, which would allow elimination of many terms and then plug in (i),(ii) and (iii). The latter (iii) can be viewed as an x projection for moving on a coulombibc potntial. Their hint says that teh partial derivative of (1/r) over dx equals -x/r3.
1.17 The operator eA has a meaning if it is expanded as a power series: eA = Sn(1/n!)An. Show that if |a> is an eigenstate of A with eigenvalue a, then it is also an eigenstate of eA. Find the latter’s eigenvalue.
Plug in |a> into the expansion for eA and evaluate the result.
1.25 The time-dependent Schrodinger equation is separable when V is independent of time. (a) Show that it is also separable when V is a function only of time and is uniform in space. (b) Solve the pair of equations. Let V(t) = V cos wt; find an expression for Y(x,t)) in terms of Y(x,0). (c) Is Y(x,t) stationary in the sense specified in Section 1.14 (i.e. Y*(x,t)Y(x,t) = y*(x)y(x))?
(a) The trick to separation of variables is.... to separate them, i.e. plug your function Y(x,t) = y(x) F(t) into the time dependent Schrodinger, divide the resulting equation by y(x)F(t) and collect all functions of x on one side while the rest (dependent on t only) - on the other side. If you succeed, the two sides can be equal only when they equal a constant (independent of x and t)
(b) When solving the two eqautions separately, note that they have the same constant (from (a)).
(c) Just evaluate your solution directly
1.21 Evaluate the quantity D4xD4p for the ground state (n = 1) of a particle of mass m in a box of length L, where D4W = <(W -<W>)4>1/4.
Follow the difinition of <W> = <y|W|y> and evaluate. You may need to refresh trigonometric relations sin2a = (1 -cos 2a)/2
1.23 The only non-zero matrix elements of x and px for a harmonic oscillator are <n+1|x|n> = (h/4pmw)1/2(n+1)1/2; <n-1|x|n> = (h/4pmw)1/2n1/2; <n+1|px|n> = i(hmw/4p)1/2(n+1)1/2; <n-1|px|n> = -i(hmw/4p)1/2n1/2 .Use the completeness relation (Ss|s><s| = 1), to deduce the value of the matrix element <n|xpx2x|n>
The suggestion means that <n|xpx2x|n> can be written as SjSlSs<n|x|l><l|px|s><s|px|j><j|x|n>. Alternatively, you can also express x and px via the lowering and raising operators, as explained in class.
2.4 A particle was prepared travelling to the
right with all momenta between (k - Dk/2)h/2p and (k + Dk/2)h/2p contributing equally to the wavepacket.
Find the explicit form of the wavepacket at t = 0, normalize it,
and estimate the range of positions, Dx,
whithin which the particle is likely to be found. Compare the
last conclusio with a prediction based on the uncertainty
principle. Hint:e
It is descriptive enough.
2.16 Calculate the energies and wavefunctions for a particle in one dimensional square well in which potential energy rises to a finite value V at each end, and is zero inside the well; that is :
V(x) = V for x < 0 and x > L
V(x) = 0 for 0 < x < L
Show that for any V and L there is always at least one bound level, and that as V --> oo the solutions coincide with those of standard PIB.
The final equation can be solved graphically
2.26 The oscillation of the atoms around their equilibrium positions in teh molecule HI can be modelled as a harmonic oscillator of mass m~mH (mass of iodine is very large) and force constant k = 313.8 N/m. Evaluate the separation of the energy levels and predict the wavelength of light needed to induce a transition between neighboring levels.
Recall the relation between the energy of stationary levels in HO and the frequency.
3.3 Calculate the probability of finding an electron within a sphere of radius ao (Bohr radius)for (a) 3s - orbital, (b) a 3p - orbital of the hydrogen atom.
The integrals need to be evaluated numerically.
3.4 Calculate the values of (a) <r> and (b) <1/r> for a 3s and 3p-orbital of a hydrogenic atom of atomic number Z.
You can inetgrate or use the relation I gave you in the lecture
P3.17 Calculate the values of (a) <r> and (b) <1/r> for a 3s and 3p-orbital of a hydrogenic atom of atomic number Z.
You can inetgrate or use the relation I gave you in the lecture
P4.2 Evaluate (a) [ly2, lx], (b) [ly2, lx2], (c) [lx,[lx,ly]
Use the basic commutators
P4.11 Calculate the matrix elements (a) <0,0|lz|0,0>, (b) <2,1|l+|2,0>, (c) <2,2|l+2|2,0>, (d) <2,0|l+l-|2,0>, (e) <2,0|l-l+|2,0>, (f) <2,0|l-2lzl+2|2,0>,
Follow the derived rules for these operators
P6.10 Calculate the first order correction to the energy of a ground state HO subject to an anharmonic perturbation ax3 + bx4, where a and b are small constants. Consider three cases: (a) bond expansion, x>0, and compression x < 0 are possible, (b) during expansion, x>0, (c) during compression.
While in evaluation for (a) you can use lowering and raising operators, (b) and (c) need more attention. There are a few different way to solve it. It seems that direct integration with recognition of the parity of the perturbation is the simplest.
P6.17 A particle of mass m is confined to a one-dimensional square x ={0,L}. Choose trial functions of the form (a) sin kx, (b) (x-x2/L) + k(x-x2/L)2 , (c) e-k(x-L/2) - e-kL/2 for x > L/2, and ek(x-L/2) - e-kL/2 for x < L/2. Find the optimum values of k and the corresponding energies.
Express the expectation value for energy <E> = <y|H|y>/<y|y> and find its minimum.
P6.5 A simple calculation of the energy of the helium atom supposes that each electron occupies the same hydrogenic 1s-orbital (but with Z=2). The electron-electron interaction is regarded as a perturbation and calculation gives: E' = (5/8)Z/(4peoao). Estimate (a) the binding energy of helium, (b) its first ionization energy.
The wavefunction for the ground state is Y=y1s(1)y1s(2) and energy, within the first pertubartion limit, E = E(0) + E', where E(0) is the energy of two electrons being in 1s orbital without taking into account their repulsion. How would you defind the binding energy then?
P6.22 An electric field in the z- direction is increased linearly from zero. What is the probability that a hydrogen atom, initially in the ground state, will be found with its electron in a 2pz-orbital at time t?
Follow the derived formula that cf(t) = -i2p/h (Int|0t H(1)fi(t) exp(iwfit) dt. Plug in the energies and the H(1)fi(t) = -<2pz|ezEz|1s>*t, where <2pz|z|1s> = gao.
E. Rec
E5.137 Li.
d.
E5.17 Li.
f.
P5.17 Regard the naphthalene molecule as having C2v symetry, which is a subgroup of its full symmetry group. Consider the p-orbitals on each carbon as a basis. What symmetry species do they span? Construct SALB basis.
Remember discussion in class, that sv(xz) goes though the longest axis of naphthalene and sv'(yz) - through the shortest. Reduce the reducible reperesentation of p10 by labeliing each p-orbital. Use projection operators to get SALC
E10.5 Express the moment of inertia of an octahedral AB6 molecule in terms of its bond lengths and the masses.
Take advantage that it is a spherical and chose the rotation axis through C4 connecting BAB.
E10.8 Which of the molecules may show a pure rotational microwave absorption spectrum :a)H2, b)CH4, c) N2O, d) H2O, e)CH2Cl2
Recall the requirement that m has to be nonzero.
E10.9 The microwave spectrum of 1H127I consists of a series of lines separated by 12.8cm-1. Compute its bond length. (m(127I) = 126.9045 amu)
Using the relation B (cm-1)= 16.8576/I(amuA2) would simplify the calculations. Obviously, it is presumed that the contribution from centrfugal distortion here is small.
E10.19 How many normal modes of vibration are there for a) H2O2, b) C6H6, c) HC#C-C#CH (# - triple bond), d) CO2?
Draw the molecules, identify their symmtery groups and reduce G3N into irreps for each. How many vibrations are IR active? Taking into account that doubly degenerate structures have the same frequency, how many IR peaks will be there?.
Identify only the fundamental stretching modes in benzene, their IR and Raman activity, as well as, approximate frequency range.
Since only stretching modes are of interest, consider only the corresponing basis of rCH and rCC.
E7.15 Li.
Draw.
E7.16 Li.
Draw.
E7.21 Li.
Draw.
E. L.
E. Fo
E. F