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PS4_PHYS3316

# PS4_PHYS3316 - Physics 3316 Spring 2010 Basics of Quantum...

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Physics 3316, Spring 2010 Basics of Quantum Mechanics Problem Set 4 (Due in lecture, Feb 22, 2010) Required Readings : French and Taylor Ch. 3, Griffiths Ch. 1 1 Propagation of wavepackets in an external potential In lecture, we considered the correspondence between particles moving in free space according to an energy function E ( p ) = p 2 2 m and wave-packets propagating under a dispersion relation given by ω ( k ) = E/ planckover2pi1 = planckover2pi1 k 2 2 m . The link between the particle’s momentum p and the wavelength λ of the wavepacket is provided by de Broglie’s relation p = h/λ = planckover2pi1 k . In this problem you will consider the same correspondence but for a particle in a system with potential energy. Consider a particle whose energy is given by H = p 2 2 m + U ( x ) . Written this way, as a function of the momentum p , the energy function of the particle has a special name. It is known as the “Hamiltonian,” and so we call the energy here “H” instead of the more familiar “E”. Imagine that an external force F ext ( t ) is applied to the particle in this system in such a way that the momentum of the particle as a function of time is given by p ( t ). a) Determine dp dt ( t ) in terms of the force acting on the particle F ext ( t ), the position of the particle x ( t ) and a derivative of the potential function V ( x ). (Note that this amounts to applying the concept of conservation of momentum.) b) Explain (briefly) why d dt H = F ext ˙ x ( t ). Thus, show that ˙ x = ∂H ∂p . (1) Is this relation true even when the external force is zero? c) Show that when the external force is zero we also have ˙ p = ∂H ∂x . (2) Eq. (1) and Eq. (2) taken together are known as the canonical equations of Hamilton.

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PS4_PHYS3316 - Physics 3316 Spring 2010 Basics of Quantum...

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