Download 1. Consider an electron moving between two atoms making up a

1. Consider an electron moving between two atoms making up a diatomic molecule. Let the state |1i correspond
to the electron being on atom 1 and state |2i correspond to its being on atom 2(Assume that |1i and |2i
are orthogonal). The Hamiltonian matrix for the lowest two states of the system can then be written, in
this basis, as
∙
¸
0 
[] =
 0
where 0 and  are positive real constants having units of energy.
(a) Find the eigenstates and eigenenergies of . Find an expansion for the basis vectors {|1i |2i} in terms
of the eigenstates of .
(b) If the electron is on atom 1 at time  = 0, find the state of the system at an arbitrary time  ≥ 0.
(c) What is the probability at time  that the electron will still (or again) be found on atom 1? On atom
2?
2. (a) Prove:
i. The eigenvalues of an anti-Hermitian operator are strictly imaginary
ii. and (ii) eigenvectors of an anti-Hermitian operator corresponding to different eigenvalues are necessarily orthogonal.
(b) Complete the following using the definition of the terms provided
i.
ii.
iii.
iv.
An
An
An
An
operator
operator
operator
operator
 is Hermitian if
 is Anti-Hermitian if:
 is Unitary if:
 is Normal if:
(c) For a single particle moving in 3D, evaluate the following expression in terms of appropriate wave
functions  () and () (You should be able to just write these down with essentially no computation.)

i. h||i
=
 · |i

ii. h|
=
 · ̂ = P3   is the
(d) If ̂ and ̂ are unit vectors decribing arbitrary directions in space, then 
=1



component
h
iof  along ̂ Similarly,  · ̂ is the component of  along ̂ Compute the comutator
 · ̂ 
 · ̂  expressing your result in terms of ̂ · ̂ For what relative orientation of ̂ and ̂ will 

and  have common eigenstates?
3. In a 2-dimensional subspace spanned by the orthonormal vectors |1i and |2i a certain linear operator 
has the following action:
|2i = |1i
|1i = −|2i
(a) Construct the matrix [] representing the operator  in this representation.
(b) Find the eigenvalues and normalized eigenvectors of 
(c) Construct unitary matrices [ ] and [ + ] that will transform from the given basis {|i}, to a new one
{| i} in which  is represented by a diagonal matrix [0 ], and explain how you would transform [].
4. Consider a particle moving in 1D only under the influence of a constant force 0 .
(a) Express the Hamiltonian for this system in terms of the 1D position and wave-vector operators  and
. [Note: You will clearly need to construct the potential energy operator for this situation.]
(b) Express the energy eigenvalue equation for this system as a differential equation in the wavevector or
momentum representation, denoting the energy eigenfunctions by  () = h| i where  = 0 
(c) Solve the differential equation to obtain acceptable momentum space eigenfunctions for this system.
Determine the allowed eigenvalues and the degeneracy of each eigenvalue.
5. Let  be a Unitary operator with eigenvalues { 0  00  · · · } and eigenvectors |i |0 i |00 i · · · 
(a) If |i is an eigenket of  , so that  |i = |i show that || = 1. (Note: if || = 1 then ∗ = −1 .)
(b) If |i is an eigenket of  , so that  |i = |i show that it is also an eigenket of  + and determine
the associated eigenvalue. [Hint consider the action of  +  on |i.]
(c) If |i and |0 i are eigenstates of  , show that if  6= 0  then h0 |i = 0 so that (since  is normal)
the eigenstates of  form an ONB.
6. A system with a Hamiltonian 0 is subject to a perturbation  .
the matrices representing these two operators are
⎛
⎞
⎛
0 0 0
0
0
⎜ 0 0 0
⎜
0
0 ⎟
⎜
⎟
⎜
⎜
⎟
0 ⎟
[] = ⎜ 0 0 0 0
[ ] = ⎜
⎜
⎝ 0 0 0 30 0 ⎠
⎝
0 0 0
0 30
In an ONB {|  i} of eigenstates of 
0 ∆ 0
0
0
∆ 0 0
0
0
0 0 0
0
0
0 0 0
0
2∆
0 0 0 −2∆
0
⎞
⎟
⎟
⎟
⎟
⎠
(a) What are the eigenvalues and degeneracies of 0 ? (Explain why you can just write these down by
looking at the matrix for .)
(b) Note:  is “block diagonal” in this basis. Draw on the two matrices above the “blocks” that this
sentence refers to. Using this fact, without doing a direct computation, deduce the operator  =
[0   ] 
(c) By working in each eigenspace  of 0 (i.e., in each block), find the eigenvalues and eigenvectors of
the operator tot = 0 +  Do 0 and  together form a Complete Set of Computing Observables’s
for this space?
7. The parity operator Π acts on the basis states |i of the position representation to ”invert them through
the origin”, i.e.,
Π|i = | − i
(a) Show that Π is both Hermitian and unitary.
(b) Determine possible eigenvalues  of any operator Λ that is both Hermitian and Unitary.
(c) Show that  + = 12 (1 + Π) and − = 12 (1 − Π) are projectors onto the eigenspace of the parity
operator, and determine the parity eigenvalues associated with those states.
(d) From an arbitrary state |i described by the wave function  ()  find the wave function that represents
the projection of |i onto the eigenspaces of Π
8. Consider a triatomic molecule with three identical atoms that are bound together with each atom at its own
corner of an equilateral triangle of edge length  In an orthonormal basis of atomic states {|i|  = 1 2 3}
the Hamiltonian is represented by the matrix
⎛
⎞
0 0 0
[] = ⎝ 0 0 0 ⎠
0 0 0
for which the following states
|1 i =
1
(|1i − 2|2i + |3i)
6
1
|2 i = √ (|1i − |3i)
2
1
|3 i = √ (|1i + |2i + |3i)
3
form an ONB of energy eigenstates (you should figure out what the energies are using a previous problem.).
Suppose at some instant of time, the electron is in state |1i when a measurement of the total energy of the
system is made.
(a) Using results from an earlier problem, determine what values can be obtained in such a measurement,
and the probability of obtaining each one of those values.
(b) Determine the probability that, as a result of the measurement, the system will be left in each of the
states |1 i |2 i and |3 i.[Hint: this is sort of trick question, so pay attention to what you say.]
9. Up till this point we have considered particles with zero spin. Now consider a particle of spin 12 (such
as an electron). Assume that the state space of such a particle is spanned by a complete ONB of states
{|  i} each of which corresponds to a particle at a specific point  ∈ 3 and in a particular “spin state”,
with the -component of spin  taking on just two possible values,  ∈ { 12  − 12 }. The state with  = 12
is said to be “spin-up”, that with  = − 12 is said to be “spin-down”. Let  be the operator such that
 |  i =  |  i.
  ] = 0
(a) Show that [
(b) Write down completeness and orthonormality relations for the ONB {|  i}. Note that these states
have both a continuous index and a discrete one, so that one has to do the correct kind of summation,
and use the correct delta function for each index.
(c) Express an arbitrary state vector |i for a spin 12 particle as an expansion in this basis, showing that it
requires two separate functions  + () and  − (). Assuming that h|i = 1 what joint normalization
condition should these two functions satisfy for a physical state of the system? What is the physical
interpretation of each function? If a measurement is made of the spin of the particle, how do we
determine from our knowledge of these two functions the probability of measuring  = 12 or −12?
10. Briefly but clearly state the four postulates of the General Formalism of Quantum Mechanics as we have
developed them in this course (not the postulates of Schrodinger’s wave mechanics for a single particle).
11. In a certain 3 dimensional quantum state space spanned by an ONB of states {|1i |2i |3i}, the spectrum
of a particular observable ̂ consists of the two values { 0  3 0 }  The eigenvalue  0 is doubly degenerate,
the value 3 0 is nondegenerate. A corresponding ONB of eigenstates of ̂ is
For eigenvalue  0
:
| 0  1i =
| 0  2i =
For eigenvalue 3 0
1
√ [|1i + |2i + |3i]
3
1
√ [|1i − |2i]
2
:
|3 0 i =
1
√ [|1i + |2i − 2|3i]
6
Suppose that the quantum system is in the state
1
|i = √ [|1i + |3i]
2
at the time of measurement of ̂
(a) What is the mean value of ̂ that will be obtained in an ensemble of such measurements?
(b) What is the probability of measuring the value 3 0 ?
(c) If the eigenvalue  0 is actually obtained in the measurement, what possible state or states could the
system actually be left in as a result of the measurement? [Hint: Be careful. Apply the postulates as
though your life depended on it.]
³
´
³
´
12. Let ̂ be an operator that is both unitary and Hermitian. Let ̂+ = 12 1 + ̂ and ̂− = 12 1 − ̂ 
(a) Evaluate the operator ̂ 2  and use your result to determine the possible eigenvalues of ̂ .
(b) Show that ̂+ and ̂− are both projection operators.
(c) Show that if |i is any state, then ̂+ |i and ̂− |i [if they are not zero], are eigenvectors of ̂  and
identify the eigenvalue for each.
(d) Onto what do ̂+ and ̂− project?
13. Prove/answer/complete the following:
(a) If |i is a normalized eigenstate of  prove that the statistical uncertainty ∆ associated with a
measurement of  on this state is zero.
(b) Show: If there exists an ONB {|   i} of simultaneous eigenstates of two observables  and  then
 and  must commute.
(c) If  is a unitary operator, and  |i = |i, prove that  + |i = ∗ |i
(d) Suppose that  and  do not commute, but that one common eigenstate | i of  and  does happen
to exist. Show that, in this case, the operator  = [ ] is not invertible.
14. In a certain triatomic molecule, the three atoms comprising the molecule arrange themselves in a straight
line
0 − − − −0 − − − −0
with equal distances between the central atom and the two atoms on either side of it. The Hamiltonian for
an extra electron bound to such a molecule can be written, in a basis of “atomic” states {|1i |2i |3i}  in
the form
⎛
⎞
0 −
0
[] = ⎝ − 0 − ⎠
0 − 0
where 0 represents the mean energy of the electron when it is sitting on any one of the three atoms, and 
is a positive real parameter representing the interaction of the electron with a neighboring atom (note that
the state |2i is associated with the electron being on the central atom). There is no interaction with atoms
more than one atom away.
(a) Find the energy levels and degeneracies associated with this system.
(b) Find an orthonormal basis of eigenstates for this system, expressed either as linear combinations of
the states .{|1i |2i |3i}  or as column vectors in that representation.
15. Let  be an operator represented in a certain ONB basis by the matrix
⎛
⎞
0 −2 0
0 ⎠
B = ⎝ 2 0
0 0
2
(i) Is  Hermitian? (ii) Is  Unitary? (iii) Find the spectrum of . (iv) Find an ONB of eigenvectors of
.