1. A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. For all parts of this question, suppose we have an initial state vector |(t = 0)) = E₂), the 2nd energy eigenstate. (This is also called "the 1st excited state", since E₁ is the lowest state or "ground state".) c) Compute (p(t)). Does your answer make physical sense? Why? d) Calculate the probability that a measurement of position will find it somewhere between L/4 and 3L/4. e) Compute Ax and Ap for the above state, and comment on their product AxAp. Feel free to use a computer for any integrals you find nasty. Recall from earlier this term how we define the standard deviation: Ax = [y]x²|y) — [y]x]y}²
1. A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. For all parts of this question, suppose we have an initial state vector |(t = 0)) = E₂), the 2nd energy eigenstate. (This is also called "the 1st excited state", since E₁ is the lowest state or "ground state".) c) Compute (p(t)). Does your answer make physical sense? Why? d) Calculate the probability that a measurement of position will find it somewhere between L/4 and 3L/4. e) Compute Ax and Ap for the above state, and comment on their product AxAp. Feel free to use a computer for any integrals you find nasty. Recall from earlier this term how we define the standard deviation: Ax = [y]x²|y) — [y]x]y}²
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![1.
A particle of mass m is in an infinite square well potential of width L, as in McIntyre's
section 5.4. For all parts of this question, suppose we have an initial state vector |(t = 0)) =
E₂), the 2nd
energy eigenstate. (This is also called "the 1st excited state", since E₁ is the lowest
state or "ground state".)
c) Compute (p(t)). Does your answer make physical sense? Why?
d) Calculate the probability that a measurement of position will find it somewhere between
L/4 and 3L/4.
e) Compute Ax and Ap for the above state, and comment on their product AxAp. Feel free to
use a computer for any integrals you find nasty. Recall from earlier this term how we define the
standard deviation:
Ax = [y]x²|y) — [y]x]y}²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0695e04c-6177-4dae-ae8e-62e62853c8ae%2F91e3de35-2a03-403f-ae86-bf1a446c70df%2F2w25ia1_processed.png&w=3840&q=75)
Transcribed Image Text:1.
A particle of mass m is in an infinite square well potential of width L, as in McIntyre's
section 5.4. For all parts of this question, suppose we have an initial state vector |(t = 0)) =
E₂), the 2nd
energy eigenstate. (This is also called "the 1st excited state", since E₁ is the lowest
state or "ground state".)
c) Compute (p(t)). Does your answer make physical sense? Why?
d) Calculate the probability that a measurement of position will find it somewhere between
L/4 and 3L/4.
e) Compute Ax and Ap for the above state, and comment on their product AxAp. Feel free to
use a computer for any integrals you find nasty. Recall from earlier this term how we define the
standard deviation:
Ax = [y]x²|y) — [y]x]y}²
Transcribed Image Text:In classical mechanics, we often solve problems by using Newton's second law F = ma to predict the
position r(t) of a particle subject to some known forces. Another common method is the energy method,
whereby we use conservation of energy and the relation E = T + V between the total energy (E)
and the kinetic (7) and potential (V) energies to predict the motion. Of course, the two methods are
related because the force is related to the potential energy by
Fx
dV
dx
(5.4)
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