Question 1
[20]
1.1 List with reason, three properties of a valid wave of a bounded state.
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1.2 Replace the following classical mechanical expressions with their corresponding
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quantum mechanical operators.
a. K.E.= ½ mv2 in three-dimensional space.
b. p = mv, a three-dimensional cartesian vector.
c. y-component of angular momentum: Lv= ZPx - xp,.
1.3. How to describe a system in quantum mechanics?
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1.4 For a particle moving freely along the x-axis, show that the Heisenberg uncertainty
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principle can be written in the alternative form: !:::.'fA:::.x 'A2./ 4n:where f:::.xis the
uncertainty in position of the particle and !:::.'iAs the simultaneous uncertainty in the
de Broglie wavelength.
1.5 What is the significance of wave packet
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Question 2
[20]
2.1 Consider a one-dimensional particle which is confined within the region 0 :=x:; :=a:; and
whose wave function is ljJ (x, t) = sin (n:x/a) exp (-iwt).
(a) Find the potential V(x).
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(b) Calculate the probability offinding the particle in the interval a/4 :=x:; :=:;3a/4.
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2.2 Consider the one-dimensional wave function
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LV(x)=A(x/xo)n e-xlxo
where A, n and xo are constants. Using Schrodinger's equation, find the potential
V(x) and energy f for which this wave function is an eigenfunction. (Assume that
as X
V(x) 0).
Question 3
[20]
3.1 The wavefunction of a particle moving in the x-dimension is
1/-(1X)
=
{
Nx(L-x)
0
0 <x < L
elsewhere
3.1.1 Normalize the wavefunction
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3.1.2 Determine the expectation value of x
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3.2 Evaluate the probability current density of the wavefunction,
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'-I'(x) = 5exp(-3ix)
3.3
The potential function V(x) of the problem is given by
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V(x)={Vo x>O
0 X< 0
where Vo is constant potential energy.
Find the wave function for E < Vo where Eis the incident particle energy and interpret
the results.
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