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0. What are the possible outcomes for measurements of A and what is the probability to obtain these outcomes in state j .t D 0/i? (c) Determine the state j .t/i of this system for times t > 0. (d) Calculate the expectation value hA i.t/ for the measurement of A for times t > 0. (e) At time t the observable A is measured. What are the probabilities for the possible outcomes of this measurement? Problem 33) Radial wavefunctions for the hydrogen atom (extra credit 6 points) The eigenvalue equation that determines the energies and radial wave functions for the hydrogen problem is „2 1 d 2 h „2 `.` C 1/ e2 1i 2me r dr 2me r2 4 "0 r R`.r/ D ER`.r/ .rR`/ C (6) with constraints rR.r / D 0 for r ! 0 and R1 R.r /2r 2dr < 1. Here, me denotes the mass and 0 e > 0 the modulus of the charge of the electron. It is customary to define the Bohr radius a0 D „2 4 "0 (7) me e2 (8) and the Rydberg ER D „2 D me e2 2 2mea02 2„2 40 (a) Transform the eigenvalue equation to u`.r/ D rR`.r/ and use the scaled energy D 2me . E/ (9) „2 . What is the form of the constraints in terms of u.r/? p (b) Introduce the dimensionless coordinate x D 2Är where Ä D and the dimensionless posi- tive real quantitiy 1 (10) D: Äa0 Use the notation u`.r/ D y`.x/ and determine the differential equation for y`.x/. (c) Use the ansatz y`.x/ D x`C1e x=2w.x/ and determine the differential equation for w.x/. Hint: The solution is xw00.x/ C .2` C 2 x/w0.x/ .` C 1 /w.x/ D 0: (11) (d) Prove that the differential equation x/˚ 0.x/ ˛˚.x/ D 0 (12) x˚ 00.x/ C . has a solution given by the following series ˛ x ˛.˛ C 1/ x2 ˚.a; bI x/ D 1 C C C::: (13) 1Š . C 1/ 2Š (This is the confluent hypergeometric function, which contains the parameters ˛ and . It is defined for parameters that are not equal to 0 or a negative integer.) (e) Solve the differential equation (11) using part (d) and argue that the series solution for w must only include a finite number of terms. Conclude that only those solutions are allowed for which D ` C 1 nr is an integer with nr D 0; 1; 2; : : : ; 1. These integer values of are denoted by n Á ; they are the principal quantum number n of the hydrogen problem. State the form of the normalizable eigenfunctions of Eq. (6) and the possible eigenvalues E. (f) Give explicit expression for the (non-normalized) wave functions for n D 1; 2; 3 and the re- spective allowed values of ` D 0; 1; 2. Hinweis: Mit Normierung, die hier nicht verlangt ist, ergibt sich s 12 .n C `/Š .2r =a0/`e r=.na0/˚ 2r Rn`.r / D pa03 n`C2.2` C 1/Š .n ` 1/Š nC`C1; 2`C2; na0 (14) ">
