Download IB Physics Quantum Physics Schrodinger, Uncertainty Principle, and

IB Physics
Quantum Physics
Schrodinger, Uncertainty Principle, and Nuclear energy levels
1.
Some of the energy levels of the hydrogen atom are shown below.
–––––––––––––– – 0.54 eV
–––––––––––––– – 0.85 eV
–––––––––––––– – 1.51 eV
–––––––––––––– – 3.39 eV
–––––––––––––– – 13.6 eV
Electrons are excited to the 0.85 eV level. How many different photon frequencies will be
observed in the emission spectrum of hydrogen?
A.
3
B.
4
C.
5
D.
6
(1)
2.
The diagram below shows some possible electron transitions between three principal energy
levels in the hydrogen atom. Which electron transition is associated with the absorption of a
photon of the longest wavelength?
0 eV
B
A
D
–1.2 eV
C
–13.6 eV
(1)
1
3.
The diagram below shows three energy levels of a certain atom.
0
T
–5.0 eV
S
–15.0 eV
The photon associated with the energy change T has frequency fT and the photon associated
with the energy change S has frequency fS
The ratio
A.
1
.
3
B.
1
.
2
C.
2.
D.
3.
fS
is
fT
(1)
4.
The graph below shows the variation with distance r from the nucleus of the square of the wave
function,  2, of a hydrogen atom according to the Schrödinger theory.
2
0
0
a
r
It may be deduced that the distance of the electron from the nucleus
A.
is most likely to be near a.
B.
is always a.
C.
is always less than a.
D.
is always greater than a.
(1)
2
5.
A beam of electrons of uniquely defined wavelength  is incident on an aperture of height d.
The beam is traveling along the x direction. The height d is of the same order as .
After passing through the aperture, the component of momentum in the x direction is px and the
component in z the direction is pz. Which of the following shows the uncertainty in px and the
uncertainty in pz?
px
pz
A.
0
0
B.
0
h
4 d
C.
D.
h
4 d
h
4 d
0
h
4 d
(1)
6.
The diagram below shows the three lowest energy levels of an atom of an element.
energy
n 3
n 2
n 1
3
Which of the following diagrams best represents the emission spectrum that results from
electron transitions between these energy levels?
B.
A.
increasing wavelength
C.
increasing wavelength
D.
increasing wavelength
increasing wavelength
(1)
7.
This question is about a model of the atom.
The Schrödinger model of the atom pictures electrons as clouds of negative charge surrounding
the nucleus. The distribution of charge and mass can be represented by a three-dimensional
standing wave.
(a)
Identify the feature of the standing wave that gives the probability of finding the electron
at a particular position.
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(2)
An electron of mass m in an atom has total energy E, potential energy EP and kinetic energy EK.
(b) Write down expressions for
(i)
the relation between E, EP and EK.
...........................................................................................................................
(1)
(ii) the momentum p of the electron in terms of EK.
...........................................................................................................................
(1)
(iii) the associated wavelength λ of the electron in terms of its total energy E.
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(2)
(Total 6 marks)
4
8.
This question is about the Bohr model of the hydrogen atom.
(a)
The diagram below shows the three lowest energy levels of a hydrogen atom as predicted
by the Bohr model.
Energy
n=3
–1.51 eV
n=2
–3.40 eV
n=1
–13.6 eV
State two physical processes by which an electron in the ground state energy level can
move to a higher energy level state.
1.
................................................................................................................................
................................................................................................................................
................................................................................................................................
2.
................................................................................................................................
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................................................................................................................................
(2)
(b)
A parallel beam of white light is directed through monatomic hydrogen gas as shown in
the diagram below. The transmitted light is analysed.
white light
beam
hydrogen
gas
transmitted light
beam
White light consists of photons that range in wavelength from approximately 400 nm for
violet to 700 nm for red light.
5
(i)
Determine that the energy of photons of light of wavelength 658 nm is about
1.89 eV.
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...........................................................................................................................
(2)
(ii)
The intensity of light of wavelength 658 nm in the direction of the transmitted
beam is greatly reduced. Using the energy level diagram in (a) explain this
observation.
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(3)
(iii) State two ways in which the Schrödinger model of the hydrogen atom differs from
that of the Bohr model.
1.
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.................................................................................................................
2.
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(2)
(Total 9 marks)
6
9.
This question is about atomic spectra and energy levels.
Diagram 1 below shows part of the emission line spectrum of atomic hydrogen. The
wavelengths of the principal lines in the visible region of the spectrum are shown.
Diagram 2 shows some of the principal energy levels of atomic hydrogen.
Diagram 1
Red (R)
656 nm
Diagram 2
Blue (B) Violet (V)
0
–0.54
–0.85
–1.5
486 nm 434 nm
–3.4
wavelength
Energy / eV
–13.6
(a)
Show, by calculation, that the energy of a photon of red light of wavelength 656 nm is 1.9
eV.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(b)
On diagram 2, draw arrows to represent
(i)
the electron transition that gives rise to the red line (label this arrow R).
(1)
(ii)
a possible electron transition that gives rise to the blue line (label this arrow B).
(1)
(Total 5 marks)
7
10.
This question is about the spectrum of atomic hydrogen and Bohr theory.
In 1890 Johannes Rydberg discovered an empirical formula that enabled the wavelengths of the
light in the atomic line spectrum of hydrogen to be calculated. The formula is
1
1 
 1
 RH  2  2 
λ
n m 
where  is the wavelength, n and m are integers and RH is the Rydberg constant.
In 1913 Niels Bohr published his theory of the hydrogen atom. This theory enabled the Rydberg
formula to be derived. His theory also showed that the energy levels En of the hydrogen atom
are given by the formula
En 
2.2 10 18
n2
where En is measured in joules.
(a)
Explain the significance, based on Bohr’s theory, of the integers n and m in the above
formulae.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
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(3)
(b) Using the formulae in (a)
(i)
estimate the wavelength of the spectral line that has the lowest value of wavelength
in the atomic hydrogen line spectrum;
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.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
(2)
8
(ii)
determine a value for the Rydberg constant.
.........................................................................................................................
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.........................................................................................................................
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(2)
(Total 7 marks)
11.
This question is about the Heisenberg uncertainty principle.
A beam of electrons is incident normally to the plane of a narrow slit as shown below.
slit
beam of electrons
x
The slit has width x equal to 0.01 mm.
As an electron passes through the slit, there is an uncertainty x in its position.
(a)
Calculate the minimum uncertainty p in the momentum of the electron.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
(2)
(b) Suggest, by reference to the original direction of the electron beam, the direction of the
component of the momentum that has the uncertainty p.
...................................................................................................................................
(1)
(Total 3 marks)
12.
This question is about the Bohr model of the hydrogen atom and the Heisenberg uncertainty
principle.
(a)
A postulate of the Bohr model of the hydrogen atom is that the electron revolves about
the proton in stable, circular orbits. State two other postulates of the Bohr model.
1 ..................................................................................................................................
..................................................................................................................................
2 ..................................................................................................................................
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(2)
9
In the nth energy state, the hydrogen atom has energy En and the electron orbits with speed vn in
an orbit of radius rn. En, rn and vn are given by the following relationships.
En = –
13 .606
eV
n2
rn = 0.0529n2 nm
vn =
2.19  10 6
m s–1
n
(b) Apply the expressions above, to hydrogen in its ground state, to determine the ground
state energy, the radius of the electron orbit and the electron speed.
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(2)
(c)
According to the Schrödinger model, the position and the speed of an electron are not
well defined. It can be assumed that the uncertainty in the position of the electron in a
hydrogen atom is equal to the radius of the electron orbit in the n = 1 state.
(i)
Apply the Heisenberg uncertainty principle to hydrogen in this state to show that
the uncertainty in the speed of the electron is approximately equal to the electron
speed as calculated in (b).
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(3)
(ii) Explain why the result in (i) above suggests that the idea of electron orbits, as used
in the Bohr model, is a poor one.
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(3)
(Total 10 marks)
10