Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Quantum Theory 1 - Home Exercise 4 1. Consider a free particle described by the wavefunction h i p p E ψ(x, t) = Aei ~ x + Be−i ~ x e−i ~ t where E = p2 2m . Find the probability current associated with this wave function, Interpret the different terms and show that if |A| = |B| the probability current vanishes. 2. Particle on a ring - Consider a particle that is free to move on a ring of circumference L. (a) Find the normalized stationary states of the system and explicitly show that they form an orthonormal basis. (b) Calculate the dispersion relation ωn (kn ) and show that ωn = ω−n . (c) Show that any linear combination of the stationary states is also a solution of (timeP dependant) Schroedinger’s equation. In the sum ψ = Cn ϕn , what condition must apply on the coefficients Cn so that ψ would be a ”legal” quantum state . (d) Given some initial condition ψ(x, 0) = f (x), where f (x) is some given normalized function, find a way to calculate the coefficients Cn . 3. Following the previous question, consider a particle on a ring that is initially in a state ψ(x, 0) = A ϕ− 2(x) + √ 2ϕ− 1(x) + ϕ0 (x) + √ 3ϕ1 (x) + √ 2ϕ2 (x) , where ϕn (x) correspond to the n0 th eigenstate you found in question (2) (a) Find |A|. (b) Find ψ(x, t). (c) We define the evenness operator be Ẑϕn (x) = 1+(−1)n ϕn (x). 2 Find the probability of having Ẑ = 1. (d) Find the expectation value for the particle’s energy hĤi, show that it is time independent. (e) Find the probability of measuring p̂ = 4π L. 1 (f) Find the probability of measuring p̂ = 2π L. At some time t0 we measure the evenness of the wavefunction and get Ẑ = 1. (g) Write down the wavefunction ψ 0 (x, t) for times t > t0 . Make sure to normalize it properly. (h) Repeat articles (d) (e) and (f) for the system at times after t0 . 2