Problem 4 Consider the Lagrangian of the one-dimensional harmonic oscillator, m L = -2 m 2 (i) Write down the Euler-Lagrange equation for the system. (ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations. (iii) Consider the following complex quantities a == mw 2 гр mw гр x + a* := x- mw. 2 mw Calculate aa* and re-express the Hamiltonian in terms of a and a*. (iv) For two given functions A(x, p) and B(x,p), the Poisson bracket {A, B} is defined as ДАӘВ {A, B} : === дх др ӘАӘВ др дх Calculate the Poisson brackets {a, a*}, {a, H}, and {a*, H}. (v) Write down the time evolution equation for a(t), and its solution.
Problem 4 Consider the Lagrangian of the one-dimensional harmonic oscillator, m L = -2 m 2 (i) Write down the Euler-Lagrange equation for the system. (ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations. (iii) Consider the following complex quantities a == mw 2 гр mw гр x + a* := x- mw. 2 mw Calculate aa* and re-express the Hamiltonian in terms of a and a*. (iv) For two given functions A(x, p) and B(x,p), the Poisson bracket {A, B} is defined as ДАӘВ {A, B} : === дх др ӘАӘВ др дх Calculate the Poisson brackets {a, a*}, {a, H}, and {a*, H}. (v) Write down the time evolution equation for a(t), and its solution.
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![Problem 4
Consider the Lagrangian of the one-dimensional harmonic oscillator,
m
L
=
-2
m
2
(i) Write down the Euler-Lagrange equation for the system.
(ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations.
(iii) Consider the following complex quantities
a ==
mw
2
гр
mw
гр
x +
a* :=
x-
mw.
2
mw
Calculate aa* and re-express the Hamiltonian in terms of a and a*.
(iv) For two given functions A(x, p) and B(x,p), the Poisson bracket {A, B} is defined as
ДАӘВ
{A, B} :
=== дх др
ӘАӘВ
др дх
Calculate the Poisson brackets {a, a*}, {a, H}, and {a*, H}.
(v) Write down the time evolution equation for a(t), and its solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F857d1b55-875c-4f26-a9e3-d336b033cff4%2F97b73d79-eb66-43af-b89e-734e6bbfa113%2F1av5cyc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4
Consider the Lagrangian of the one-dimensional harmonic oscillator,
m
L
=
-2
m
2
(i) Write down the Euler-Lagrange equation for the system.
(ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations.
(iii) Consider the following complex quantities
a ==
mw
2
гр
mw
гр
x +
a* :=
x-
mw.
2
mw
Calculate aa* and re-express the Hamiltonian in terms of a and a*.
(iv) For two given functions A(x, p) and B(x,p), the Poisson bracket {A, B} is defined as
ДАӘВ
{A, B} :
=== дх др
ӘАӘВ
др дх
Calculate the Poisson brackets {a, a*}, {a, H}, and {a*, H}.
(v) Write down the time evolution equation for a(t), and its solution.
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