PHY3063
Spring 2015
HOMEWORK E
Instructor: Yoonseok Lee
HW 1: Tipler 5-5
HW 2: Tipler 5-7.
HW 3: Tipler 5-9.
HW 4: Tipler 5-17.
HW 5: Tipler 5-24. |Ψ(x)|2 is the probability density that should be properly normalized.
HW 6: Tipler 5-25. |Ψ(x)|2 is the probability density and |Ψ(x)|2 dx is the probability.
HW 7: Tipler 5-26.
HW 8: Tipler 5-36.
HW 9: Tipler 5-46. You get the quantized wavelength λn from the standing wave condition
and then get the momentum (de Broglie) and the non-relativistic kinetic energy.
HW 10: Tipler 5-49.
HW 11: One can estimate the zero-point energy of a quantum harmonic oscillator using the
uncertainty principle. Consider a harmonic oscillator whose energy is given by
E=
p2
1
+ mω 2 x2 .
2m 2
Here we use the conventional notation for the mass (m) and natural frequency (ω) of the
oscillator. Classically, the ground state energy (minimum energy) is zero meaning that the
particle is frozen at the bottom of the potential. In quantum mechanics, the uncertainty
principle forbids this and allow an uncertainty in position as ∆x = δ. Then the momentum
uncertainty would be coming from ∆x∆p ∼ ~. For the lowest momentum one can argue
that p ≈ ∆p (why?). You can then write the total energy of the ground state in terms of δ
and minimize E(δ) to estimate the lowest energy. Follow this procedure to extract painlessly
the ground state energy (zero-point energy) of an harmonic oscillator.
1