A good example of time evolution of an operator is the position in x in 1-dimension. This simplified case allows us to deal with the wave function as just a scalar so, < ¥\â\¥ >= ƒ ¥¹ â¥dx da In classical physics, we define momentum as px = mv₂ = m. 8² Using the spatial representation in 1D, momentum is p = -ithe free particle Hamiltonian is, H = 2m მე2 Calculate the time derivative of and use the Schrodinger's Equation id/dt = Hy for a free particle to derive the quantum momentum as follow, =< P > m d<â> dt
A good example of time evolution of an operator is the position in x in 1-dimension. This simplified case allows us to deal with the wave function as just a scalar so, < ¥\â\¥ >= ƒ ¥¹ â¥dx da In classical physics, we define momentum as px = mv₂ = m. 8² Using the spatial representation in 1D, momentum is p = -ithe free particle Hamiltonian is, H = 2m მე2 Calculate the time derivative of and use the Schrodinger's Equation id/dt = Hy for a free particle to derive the quantum momentum as follow, =< P > m d<â> dt
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Transcribed Image Text:A good example of time evolution of an operator is the position in x in 1-dimension. This simplified case allows
us to deal with the wave function as just a scalar so,
< ¥\â\¥ >= ƒ ¥¹ â¥dx
In classical physics, we define momentum as pa
m
dx
dt
-
-1 8²
2m əx²
Using the spatial representation in 1D, momentum is p = -2
-ithe free particle Hamiltonian is, H
Calculate the time derivative of <x> and use the Schrodinger's Equation id/dt = Hy for a free particle to
derive the quantum momentum as follow,
=< p >
d<â>
dt
mvx = m.
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