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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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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?
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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Thats for question a when t=0. Is it the same when t=t' in question c?

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