Question

a) The wave-functions of the states [) and (o) are given by y(x) and (x), respectively. Derive the expression for the inner product (4) in terms of the wave- functions Q(x) and (x). What is the physical meaning of y(x) and (x)/2? b) Fig. 1 shows a sketch of y(x). Sketch y(x) such that the states [4) and (o) are orthogonal: (14) = 0. (x) M Figure 1 c) Assume a particle has a wave-function y(x) sketched in Fig. 2. Sketch the wave- function (p) of this particle in momentum representation. (x) Figure 2 d) Consider the state |p) which is the eigenstate of the momentum operator p such that plp) = plp) (eq 1). We also know that if the state ) has a wave-function (x) then the wave-function of 10) = ply) is $(x) = -ihy(x)/x (eq 2). Use (eq 1) and (eq 2) to derive an explicit form of the wave-function of p) in the coordinate representation. Neglect the normalization constant.

Answer 1

(a) The of the states 46x) and are Wavefuncūns 012 given (a) The wavifunclions of the slátis 193 and 10% are given as 4/4) and ple) respectively. In the Hilbert space where the scolars are complex numbers, the inner product can be written as 2016 ols Braket notation . It is defined as T8147= to p + (x) 41 x) dx where 874) is the complie conjugate of Pixi The notation inner product 30147 is also known as The wave function 414) in quantum physics gives the complex - volud probability probabilitiki and other measurements can be derived from this st gives the entire descriplīzine of 141x)/2 probobility density that particle recipes pertizuin Position. amplilade , Arit the the quantum system gives the

16) The Sleutch of 11x) is sholun below. 41x) a For onthogonality of 412) to pla): The lio wavefunctions are said to be orthogonal if jpt. 8*4 dx = 0 or <$147 =0 The combination of ever and odd funcīón gives orthogonal wanfunctiix Since the given wanfunction is ever other wavefunction P1x) must be odd an even I the as shown below. W xy/ (C) The wavefunction of a particle in coordenia representation is shown below 478) x

+♡ It ha de 4x) e 21hd In momentum space, 410) - |2ith Let 411) - constant (as shown) ii 41p) - die . = V2th tas I c.e-ipx/h Further, momenlien þ- -iħ24/2) Әк sence & 41x) = constant 24 (x) = 0 : pe corslantion 2x 4 (P) d) [py stali is the eigenstali of momilum operatón È such thet > = P1BY 10>= 147 -it 241x) ---101 dk. Thus $ 412) = or dylx) ips P1x) olny lu) = ipe in 241x) du: h 4116) = A e ipu/hte Where A= rurmaliiatin conulart