At time t = t'> 0, what is the probability that a measurement of total en- ergy will yield: i. E = E₁, ii. E = E₂, iii. a value E other than E₁ or E₂?
At time t = t'> 0, what is the probability that a measurement of total en- ergy will yield: i. E = E₁, ii. E = E₂, iii. a value E other than E₁ or E₂?
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i only need help with 2 c
Transcribed Image Text:2. In lecture we learned that if (r) is an eigenfunction of the time-independent Schrödinger
equation with eigenvalue En, then
Vn(x, t) = n(r)e iEnt/h
(5)
is a solution to the time-dependent Schrödinger equation. Now suppose that at time t = 0 we
prepare the system in the following normalized superposition state:
y(x) = c₁₁(x) + €₂₂(x)
(6)
where ₁(x), ₂(x) are eigenfunctions of the time-independent Schrödinger equation with eigenvalues
E₁, E2, respectively.
(a) At time t = 0, what is the probability that a measurement of total energy will
yield: i. E = E₁, ii. E = E₂, iii. a value E other than E₁ or E₂?
(b)
Show that
iExt/h
V(x, t) = c₁v₁(r)e
+ €2V/2(x) e
is a solution of the time-dependent Schrödinger equation.
iE₂t/h
(7)
(c) At time t = t'> 0, what is the probability that a measurement of total en-
ergy will yield: i. E = E₁, ii. E = E2, iii. a value E other than E₁ or E₂?
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Follow-up Question
Thats for question a when t=0. Is it the same when t=t' in question c?
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