aψ(x) =√^12 afor−a < x < a, otherwiseψ(x) = 0.bψ(x) = (απ)(^14)
e−αx
(^2) / 2
c ψ(x) =δ(x−x 0 )
- Use the Heisenberg uncertainty principle to estimate the groundstate energy for a particle of
massmin the potentialV(x) =^12 kx^2.
6.*Find the one dimensional wave function in position spaceψ(x) that corresponds toφ(p) =
δ(p−p 0 ).7.*Find the one dimensional wave function in position spaceψ(x) that corresponds toφ(p) =
√^1
2 bfor−b < p < b, andφ(p) = 0 otherwise.
8.*Assume that a particle is localized such thatψ(x) =√^1 afor 0< x < aand thatψ(x) = 0
elsewhere. What is the probability for the particle to have a momentum betweenpandp+dp?- A beam of photons of momentumpis incident upon a slit of widtha. The resulting diffraction
pattern is viewed on screen which is a distancedfrom the slit. Use theuncertainty principle
to estimate the width of the central maximum of the diffraction pattern in terms of the variables
given.
10.*The wave-function of a particle in position space is given byψ(x) =δ(x−a). Find the
wave-function inmomentum space. Is the state correctly normalized? Explain why.
11.*A particle is in the stateψ(x) =Ae−αx
(^2) / 2
. What is the probability for the particle to have
a momentum betweenpandp+dp?
12. A hydrogen atom has the potentialV(r) =−e
2
r. Use the uncertainty principle to estimate the
ground state energy.13.*Assume thatφ(p) =√^12 afor|p|< aandφ(p) = 0 elsewhere. What isψ(x)? What is the
probability to find the particle betweenxandx+dx?
- The hydrogen atom is made up of a proton and an electron boundtogether by the Coulomb
potential,V(r) =−e
2
r. It is also possible to make a hydrogen-like atom from a proton and
a muon. The force binding the muon to the proton is identical to thatfor the electron but
the muon’s mass is 106 MeV/c^2. Use the uncertainty principle to estimate the energy and the
radius of the ground state of muonic hydrogen.15.*Given the following one dimensional probability amplitudes in the momentum representa-
tion, compute the probability amplitude in the position representation,ψ(x). Show that the
uncertainty principle is satisfied.
(a)ψ ̄(p) =√^12 afor−a < p < a,ψ ̄(p) = 0 elsewhere.
(b)ψ ̄(p) =δ(p−p 0 )
(c)ψ ̄(p) = (απ)(^14)
e−αp
(^2) / 2
16.*Assume thatφ(p) =δ(p−p 0 ). What isψ(x)? What is< p^2 >? What is< x^2 >?
17.