Question

II(20pts). Short Problems

a) The lowest energy of a particle in an infinite one-dimensional
potential well is 4.0 eV. If the width of the well is doubled, what
is its lowest energy?

b) Find the distance of closest approach of a 16.0-Mev alpha particle incident on a gold foil.

c) The transition from the first excited state to the ground state in potassium results in the emission of a photon with = 310 nm. If the potassium vapor is used in a Frank-Hertz experiment, at what voltage would you expect to see the first decrease in current?

d) Alpha particles of kinetic energy 5.00 MeV are scattered at 90◦ by a gold foil. What is the impact parameter??

e) the lifetime of an electron at an excited state is about t 10-8s (the uncertainty of the time). What is the minimum uncertainty of the energy of the emitted photon?

III(12pts). (a) The radius of the n=1 orbit in the hydrogen atom is aB=0.053 nm. Compute the radius of the n=10 orbit.

(b) For a Hydrogen-like ion Ne9+ (atomic number Z=10), What is the radius of the n=1 and n=10 orbits?

(c) What energy E is needed to excite the electron from its ground state to its 2nd excited state for this Hydrogen-like ion Ne9+?

IV(12pts) An electron is in an angular momentum state with l= 2. (a) What is the length of the electron’s angular momentum vector?

(b) How many different possible z components can the angular momentum vector have? List the possible z components of Lz.

⃗

(c) What is the smallest angle that the ? vector makes with the z
axis?

V(12pts) The wavefunction of a particle in a 1D rigid box of length a is:

b) What is the probability of finding the particle in the interval between x=0.5a and x=0.51a?

c. Find the expectation values of <x> for the particle in the ground state.

(x,t)) = 2 sin(nx x)e−i(En / )t aa?

.? =1,2,3,...

a) Find the probability density of the particle in the ground state
|?1|2.

VI (12 pts). The hydrogen atom wave functions are n,l,ml (r,,) = Rn,l (r)l,ml ()ml () . Where n=1,2,3...; l=0,1,2,...,n-1; m =l, l-1,...,0,...-l−1, −l....-1;ms=1/2,-1/2. E =−ER

l n n2

a) Considering electron spin, what is the degeneracy of the n=1,
n=2, and n=3 energy levels?

b) How many different set (please list them) of quantum numbers (n, l, ml, ms) are possible for the n=2 level?

2−r

c) For the ground state hydrogen atom: R (r) = e aB , Write down
the expression of the radial

1,0 a3/2 B

probability of finding the electron from r =0 to r = aB .

VII(12pts). A quantum particle of mass M moves freely in a one-dimensional rigid box of length 2b. Find the allowed energies of the particle in the box and the normalized wave functions by solving the one-dimensional time-independent Schrödinger equation for

?(?) = {0, ??? 0 ≤ ? ≤ 2? ∞, ????<0????>2?

Answer #1

4.
An electron is trapped in a one-dimensional infinite potential well
of width L.
(1) Find wavefunction ψn(x) under assumption that the
wavefunction in 1 dimensional box whose potential energy U is 0 (0≤
z ≤L) is normalized
(2) Find eighenvalue En of electron
(3) If the yellow light (580 nm) can excite the elctron from
n=1 to n=2 state, what is the width (L) of petential well?

An electron is trapped in an infinite one-dimensional well of
width = L. The ground state energy for this electron is 3.8
eV.
a) Calculated energy of the 1st excited state.
b) What is the wavelength of the photon emitted between 1st
excited state and ground states?
c) If the width of the well is doubled to 2L and mass is halved
to m/2, what is the new 3nd state energy?
d) What is the photon energy emitted from the...

A particle is confined to the one-dimensional infinite potential
well of width L. If the particle is in the
n=2 state, what is its probability of detection between a) x=0, and
x=L/4; b) x=L/4, and x=3L/4; c) x=3L/4,
and x=L? Hint: You can double check your answer if you calculate
the total probability of the particle being
trapped in the well.
Please answer as soon as possible.

A. Consider a hydrogen atom with one electron and quantized
energy levels. The lowest energy level (n = 1) is the ground state,
with energy -13.6 eV. There are four states corresponding to the
next lowest energy (n = 2), each with energy-3.4 eV. For the
questions below, consider one of these four states, called one of
the first excited states.
2. Assume that this hydrogen atom is present in a gas at room
temperature (T ~ 300 K, kBT...

Take the potential energy of a hydrogen atom to be zero for
infinite separation of the electron and proton. Then the ground
state energy of a hydrogen atom is –13.6 eV. The energy of the
first excited state is:
A) 0eV
B) –3.4 eV
C) –6.8 eV
D) –10.2 eV
E) –27 eV

A particle is confined to the one-dimensional infinite potential
well of the figure. If the particle is in its ground state, what is
the probability of detection between x = 0.20L
and x = 0.65L?

Suppose that an electron trapped in a one-dimensional infinite
well of width 0.341 nm is excited from its first excited state to
the state with n = 5.
1 What energy must be transferred to the electron for this
quantum jump?
2 The electron then de-excites back to its ground state by
emitting light. In the various possible ways it can do this, what
is the shortest wavelengths that can be emitted?
3 What is the second shortest?
4 What...

Exercise
3. Consider a particle with mass m in a
two-dimensional infinite well of length L, x, y
∈ [0, L]. There is a weak potential in the well
given by
V (x,
y) = V0L2δ(x −
x0)δ(y − y0)
.
Evaluate the first order correction to the energy of the ground
state.
Evaluate the first order corrections to the energy of the first
excited states for x0 =y0 = L/4.
For the first excited states, find the points...

If an electron is confined to one-dimensional motion
between two infinite potential walls which are separated by a
distance equal to Bohr radius, calculate energies of the three
lowest states of motion.Calculate numerical value of ground state
energy and compare it with hydrogen atom ground state energy.

For a particle trapped in a one-dimensional infinite square well
potential of length ?, find the probability that the particle is in
its ground state is in
a) The left third of the box: 0 ≤ ? ≤ ?/3
b) The middle third of the box: ?/3 ≤ ? ≤ 2?/3
c) The right third of the box: 2?/3 ≤ ? ≤ L
After doing parts a), b), and c):
d) Calculate the sum of the probabilities you got for...

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