Question

Questions 1 - 5 deal with a particle in a one-dimensional infinite square well of width a where 0, 0 SX Sa V(x) = 100, Otherwise. The stationary states are Pn(x) = sin(**) with energies En = "forn = 1,2,3.. Question 1 (14 pts) Which of the following is correct? A. The Hilbert space for this system is one dimensional. B. The energy eigenstates of the system form a ID Hilbert space. C. Both A and B are correct. D. None of the above. Question 2 (14 pts) The wavefunction at time to is UN um 17 12). If a measurement of the energy is made which of the following is correct? A. The measurement of energy will be * E. B. The spatial part of the normalized wavefunction after the measurement is either 11) or 12). ! C. The spatial part of the normalized wavefunction after the measurement is either (1) or (2) D. Both A and B. E. None of the above. Question 3 (14 pts) The wavefunction at time is 12). If a measurement of the energy is made what result can we obtain? A. The measurement of energy will be un IN V B. The measurement of energy will be either C. The measurement of energy will be either Ei or Ej . D. None of the above. Question 4 (14 pts) Suppose the initial state of the particle is 03-107. What is the expectation value of the energy (E)? A. 1 + JE B. E1 or E2 c. & E 22 D. Any of En, n=1,2,3.... E. None of the above

Answer 1

A Pantele in leo infinite well of width ă are Ving= no osnza { as otherwise Yn (m) = 1 i sin (max) & E. n2th n=1,2,3... ama² ① Acconding to postulates of quantum mechanics, The enenzy eigen state is form of the Hilbent space. ie. Hilbent space (H) is a linean space, has a defined scalan Product that strictly positive. Hiis selanable and a complete state. - H is consists of a set of vectons a scalane - 5o all the properties satisfied by 4 14). The 414) given in InD Hilbent space. Thus Both A & B are connect

* At t=0 [yemina E 1) + 12) Now Probabilities. fone 11 Steure P = this kilp)12 P = fall @ 112 r 123 12 for 127 state, P = kaly) 12 h = 1 221 P 11 + 8 12) /? P2 = 1 1315 22/2) 1² = 3 Thus enengy measunent. is (E) = P Es + BELEE+ Use En anrlah n=1,2,3... gma? g eomething Ga ya (n=2) gmc MA : TEX: O V EN

And the i spatial part 117 on Both A & B of the normalized or 12% are connent - 6 (ans). They u See (Q-2 (€) Er tier measunemout of enengy. for individual measurement of enengy, we can obtain either [2/3 E, or (E ( 17 E) . phus | Option- A is connect - G If yra ħ 16,7 -1 182) Flow Probabilities : Pi = 1 <1 %)/2 = 1 Pi = 1 2/4372 - 612 Thus expectation value of enengy LES = Piet Pz Ez -45, + Ei sent = E is 2 [ (E)= a + 2 2 - 0