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Problems and Multiple Choice Questions
1.
A momentum operator in one dimension is
2.
a)
; b)
; c)
; d)
; e)
A position operator in 3 dimensions is
a)
; b)
; c)
A kinetic energy operator in 1 dimension is
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
;
d)
;
e)
a)
; b)
; c)
; d)
; e)
If two operator commute,
a) they have the same eigenvalue
b) they are not quantum mechanical operators
c) they have the same eigenfunction
d) both of them are time dependent operators
e) both of them are position dependent operators
If two operator commute,
a) they have the same eigenvalue
b) their eigenvalues can be measured simultaneously
c) their eigenvalues can not be measured
d) one of them is always zero
e) their eigenvalues always zero
If two operator commute
a) they are not Hermitian
b) there is an uncertainty between them
c) one of them is Hamiltonian operator
d) one of them is energy operator
e) they are always Hermitian
Commutation relation
of the operators
is
a) 1; b) 2; c) -2; d) -1; e) 0
Consider the operators A, B and C: [A, [ B, C] + [B, [ C, A] + [C, [ A, B] = 0
a) True; b) False
Uncertainty
between the the operators
is
a) 1; b) 2; c) -2; d) -1; e) 0
Which of the following is eigenfunction of the momentum operator
a)
; b)
; c) ; d) ; e)
Which of the following is eigenfunction of the kinetic energy operator
a)
; b)
; c) ; d) ; e)
Indicate which of the following wave functions are “acceptable.”
a) ψ =x
b) ψ =x2
c) ψ =sin x
d) ψ =exp(−x)
e) ψ =exp(−x2)
Commutation between momentum and kinetic energy operators is
a) 1; b) 2; c) -2; d) -1; e) 0
For a particle in the infinite well of width a normalized function is
a)
;
b)
; c)
; d)
;
e)
15. The full Schrödinger equation in compact form is
a)
; b)
; c)
; d)
; e)
16. General form of the time independent Schrödinger equation in 1 dimension is
a)
; b)
; c)
;
d)
;
e)
17. A physical requirement on wavefunctions is that they should be:
a) Reliable; b) square integrable; c) Zero; d) Constant; e) Discrete
18. Which of the following is not a physical requirement for a wave valid wave function?
a) single valued; b) continuous in a given region; c) symmetric; d) differentiable; e) square integrable
is solution of Schrödinger equation with zero
19. For which values of the energy the wave function
potential
a)
; b)
;
c)
;
d)
; e)
20. An electron is in an infinite square well that is 8.9-nm wide. The ground state energy of the electron is closest to:
a) 0.0066 eV
b) 0.0085 eV
c) 0.0057 eV
d) 0.0076 eV
e) 0.0047 eV
21. An electron is in an infinite square well that is 9.6-nm wide. The electron makes the transition from the n = 14 to the n =
11 state. The wavelength of the emitted photon is closest to:
a) 3400 nm; b) 4100 nm; c) 2800 nm; d) 4700 nm; e) 5300 nm
22. An electron is in the ground state (lowest energy level) of an infinite well where its energy is 5.0 eV. In the next higher
level, its energy would be closest to:
a) 5 eV; b) 20 eV; c) 12.5 eV; d) 25 eV; e) 5/4 eV
23. An electron is in an infinite square well that is 9.6-nm wide. The electron makes the transition from the n = 14 to the n =
11 state. The wavelength of the emitted photon is closest to:
a) 3400 nm b) 4100 nm c) 2800 nm d) 4700 nm e) 5300 nm
24. Conjugate of the wave function
is
a)
; b)
; c)
; d)
; e)
25. It is found that a particle in a one-dimensional box of length L can be excited from the n=1 to the n=2 state by light of
frequency ν. If the box length is doubled, the frequency needed to produce the n=1 to n=2 transition becomes
a) ν/4
b) ν/2
c) 2ν
d) 4ν
e) None of the above is correct.
26. The wave function for a particle must be normalizable because
a) the particle’s charge must be conserved.;
b) the particle’s momentum must be conserved.
c) the particle cannot be in two places at the same time.
d) the particle’s angular momentum must be conserved.
e) the particle must be somewhere.
27. Heisenberg’s Uncertainty Principle states:
a) The more precise a particle’s energy can be measured, the less precise its position can be measured.
b) A particle’s position can be measured exactly.
c) A particle’s energy can be measured exactly.
d) The more precise a particle’s momentum can be measured, the less precise its position can be measured.
e) The more precise a particle’s momentum can be measured, the less precise its energy can be measured.
28. If the uncertainty in the veocity of a proton accelerated in a laboratory is 400 m/s, that of its position is:
a) 7.88 nm ; b) 9.70 nm ; c) 112 nm ; d) 21.5 nm; e) 2.8 nm.
29. Expectation (average values of momentum operator can be written as:
a)
;
b)
;
c)
; d)
;
e)
30. Expectation values of the kinetic energy of a particle described by a wave function
in the infinite
well is
a)
; b) cannot be determined; c) 2/a; d) ; e)
31. Expectation values of
of ground state harmonic oscillator is given by
Calculate uncertainty between position and momentum.
Hint: Use the variance of measurement
32. Expectation values of
and uncertainty
.
of a particle in the infinite well box of width a is given by
Where
is quantum number
Calculate uncertainty between momentum and position. What is the
minimum uncertainty between position and momentum? Is Heisenberg uncertainty relation satisfied?
33. The wave function for a particle confined in an infinite square well arranged to the coordinate system
is given by:
where
, E corresponds to energy, m is mass of the particle, A is constants.
is Planck's constant.
a) Calculate A. (Hint: use normalization and
b) Is
valid wave function? Explain.
)
Verify that
satisfies Schrödinger’s equation
that for the region inside the well
.) What is the values of E?
d) Calculate uncertainty between x and p, by using variance of x and p.
e) What is the probability of finding particle beween
,
Hint:
c)
34. Show that for
., for some certain values of energy E. (Note
, wave function
satisfy the solution of the Schrödinger equation:
35. For the values
Schrödinger equation:
, determine potential
such that
is the solution of the
Where m is mass and Z and q are constants.
36. (Griffiths 1.4) At time t=0 a particle is represented by the wave function:
where A, a and b are constants.
a) Normalize (That is, find A, in terms of a and b).
b) Sketch
as a function of x.
c) Where is the particle most likely to be found at t=0?
d)What is the probability of finding the particle to left of a? Check your result in the limiting cases
e) What is the expectation value of ?
Solution
a) Using normalization condition:
and
b) Graph of the function:
c) To obtain most probable point sketch graph
then you the maximum point
is also most probable point.
.
d)
e) Expectation value of an operator:
37. (Griffiths Problem 1.17). A particle is represented (at time t=0) by the wave function
a) Determine the normalization constant A.
b) What is the expectation value of x?
c) What is the expectation value of momentum p?
d) Find the uncertainty in x.
e) Find the uncertainty in p.
f) Check your results are consistent with the uncertainty principle.
Solution
a) Again using the relation
b) Expectation value of x:
c) Expectation value of p:
d) Uncertainty in x:
e) Uncertainty in p:
f) Uncertainty relation:
38. A particle in the harmonic oscillator potential starts out with superposed the wavefunction:
a)
Find
.
39.
a)
b)
c)
Wave Function
What are the fundamental behaviors of the wave function?
How can you determine wave function of a quantum particle
What is the expectation value and eigenvalue of an operator? When they are the same?
a) Fundamental behaviors of the wave function
A wave function
must be
Single valued: A single-valued function is function that, for each point in the domain, has a unique value in the range.
Continuous: The function has finite value at any point in the given space.
Differentiable: Derivative of wave function is related to the flow of the particles.
Square integrable: The wave function contains information about where the particle is located, its square being probability
density. Therefore
b) The wave function can be determined by solving Schrödinger equation:
c) Expectation value of an operator is defined as
value and eigenvalues of the operator is the same.
.When
is eigen function of
40. Under what condition the function
is eigenfunction of the operator
momentum operator, if its eigenvalue is twice of the constant potential .
Solution
where
then expectation
is
41. Show that the function
is eigenfunction of momentum and kinetic energy operators.
42. Generalized uncertainty between two observable is given by:
Calculate the uncertainty relation between the observables, momentum, energy, position and time.
43. Show that, if the eigenvalue equation
then expectation value of the operator is equal to eigenvalues of the operator.
Solution
a)
Momentum operator is given by
With the eigenvalue
; the eigenvalue equation can be written as
.
Similarly kinetic energy operator can be written as:
With the eigenvalues
, then we obtain:
we can write eigenvalue equation
.
b)
Using the operator relations of the observables
relation:
, we obtain the commutation
The uncertainties are
c)
Expectation value can be defined as:
obtain:
substituting
and using normalization condition we
43. A particle in the harmonic oscillator potential starts out in the state
a) Determine A.
Solution1
a)Using the normalization conditions:
44. Find the eigenfunctions of the operators
.
Solution.
Eigenvalue equation can be written as
where is operator
Then
Integrating both sides yields
Similarly
Solution of the differential equation yields
is its eigenvalue and
is eigenfunction of .