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2. Repeat the above question with a Hamiltonian of the form H(p, T) : +V(#), 2m where p(t) and (t) are now vector functions in 3-dimensional space. (Given what you have already done, this should be very easy.)

2. Repeat the above question with a Hamiltonian of the form H(p, T) : +V(#), 2m where p(t) and (t) are now vector functions in 3-dimensional space. (Given what you have already done, this should be very easy.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Consider a Hamiltonian function of the form p2 H(p, x) + V(τ). 2m Here p and x are one-dimensional. (a) Write down Hamilton's equations. Physically interpret p, m, V for this particular system. (b) Using Hamilton's equations, solve for p as a function of m and i. (c) Rewrite the quantity p2 T= 2m as a function of m and i, and hence provide a physical interpre- tation for this quantity. 2. Repeat the above question with a Hamiltonian of the form H(p, T) = + V(7), 2m where p(t) and 7(t) are now vector functions in 3-dimensional space. (Given what you have already done, this should be very easy.) Hints: p = (Pa, Py, Pz); Ë = (x, y, z); 7² = p·p = p + p, + p?.
Transcribed Image Text:1. Consider a Hamiltonian function of the form p2 H(p, x) + V(τ). 2m Here p and x are one-dimensional. (a) Write down Hamilton's equations. Physically interpret p, m, V for this particular system. (b) Using Hamilton's equations, solve for p as a function of m and i. (c) Rewrite the quantity p2 T= 2m as a function of m and i, and hence provide a physical interpre- tation for this quantity. 2. Repeat the above question with a Hamiltonian of the form H(p, T) = + V(7), 2m where p(t) and 7(t) are now vector functions in 3-dimensional space. (Given what you have already done, this should be very easy.) Hints: p = (Pa, Py, Pz); Ë = (x, y, z); 7² = p·p = p + p, + p?.
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