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Download Quantum Theory 1 - Class Exercise 4
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Quantum Theory 1 - Class Exercise 4 1. Consider a Hamiltonian which describes a one dimensional system of two particles of masses m1 and m2 moving in a potential that depends only on the distance between them. Ĥ = p21 p2 + 2 + V (x1 − x2 ) 2m1 2m2 (a) Write down Schroedinger’s equation using the new variables X= m1 x1 + m2 x2 m1 + m2 , x = x1 − x2 . Interpret the meaning of these variables. (b) Use seperation of variables to find the equations of motion of X and x. 2. Consider a ”quantum dice”, which is just a quantum description of a regular playing dice. We define the number operator N̂ ϕn = nϕn and the evenness operator Ẑϕn = 1+(−1)n ϕn . 2 (a) When measuring the number operator, what is the probability of getting N̂ = 4? (b) When measuring the evenness operator, what is the probability of getting Ẑ = 1? (c) We measure the number operator and get N̂ = 4. Afterwards, we measure the evenness, what is the value we are going to get? (d) We measure the evenness operator and get Ẑ = 1. Afterwards, we measure the number operator. What is the probability of measuring n = 4? 3. Consider a free particle in an infinite well of length L. Find stationary states, eigenenergies, and eigenvalues of the momentum operator. 4. Consider a free particle in an infinite well of length L. At time t = 0 the particle is prepared in the state √ √ √ ψ(x, 0) = A ϕ1 + 2ϕ2 + 3ϕ3 + 2ϕ4 + ϕ5 (a) Find |A|. (b) Find ψ(x, t). (c) What is the probability of measuring an energy larger then 1 2~2 π 2 mL2 (d) What is the probability of measuring a momentum of 4π~ L (e) In a measurement of the particle’s energy we got the value we measured the particles momentum. • What is the probability of having p = 2π~ L ? • What is the probability of having p = 4π~ L ? 2 4~2 π 2 . mL2 After this measurement,