Consider a classical particle of mass m moving in one spatial dimension with position x and momentum p. The Hamiltonian of the system is p² 1 H(x,p) + k 2m 2kx² where k is a positive real parameter. (a) Find the expression of the canonical classical partition function z(B) for this system. (b) Show that for any natural integer ʼn the canonical expectation value (x²¹) satisfies [hint: you may use the formula sheet] В n (x2²n), = ² (n + 2) (²)", √π Bk) where I'(x) is the Euler Gamma function. =
Consider a classical particle of mass m moving in one spatial dimension with position x and momentum p. The Hamiltonian of the system is p² 1 H(x,p) + k 2m 2kx² where k is a positive real parameter. (a) Find the expression of the canonical classical partition function z(B) for this system. (b) Show that for any natural integer ʼn the canonical expectation value (x²¹) satisfies [hint: you may use the formula sheet] В n (x2²n), = ² (n + 2) (²)", √π Bk) where I'(x) is the Euler Gamma function. =
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![Consider a classical particle of mass m moving in one spatial dimension with
position x and momentum p. The Hamiltonian of the system is
p² 1
H(x,p)
+= kx²,
2m 2
where k is a positive real parameter.
(a) Find the expression of the canonical classical partition function z(B) for this
system.
(b) Show that for any natural integer ʼn the canonical expectation value (x²¹)
satisfies [hint: you may use the formula sheet]
r(n + 2 ) ( 2² ) ².
n
(x²n)B
=
√π
where I'(x) is the Euler Gamma function.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59e56c88-ee2e-4d60-90cb-58b6c6618de4%2F81e8aa61-6c0b-4044-8364-8d509fba162c%2Fvq8enhc_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a classical particle of mass m moving in one spatial dimension with
position x and momentum p. The Hamiltonian of the system is
p² 1
H(x,p)
+= kx²,
2m 2
where k is a positive real parameter.
(a) Find the expression of the canonical classical partition function z(B) for this
system.
(b) Show that for any natural integer ʼn the canonical expectation value (x²¹)
satisfies [hint: you may use the formula sheet]
r(n + 2 ) ( 2² ) ².
n
(x²n)B
=
√π
where I'(x) is the Euler Gamma function.
=
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