Question

I am really struggling with quantum. Can someone please help me with those questions

4.2. Using basic quantum concepts Pr 4.3 This problem reinforces your understanding of normalization, prob- ability densities, and mean (expectation) values. The ground state wave function for a particle in a wire is = (2/a)/2 sin(x/a). (a) Define the ground state wave function for the particle in a wire using Maple. Hint: see Appendix A. (b) Write down a mathematical expression for the normalization of the ground state wave function for the particle in a wire. Use Maple to show that is normalized. Hint: Do not forget to use the correct limits of integration. (c) Recall that v2 dx is the probability for finding the particle in an in- finitesimal neighborhood of. Hence 22 dx is the probability for finding the particle between the points I and 3. Use Maple to cal- culate, for the ground state, the probability of finding the particle in (i) the middle third of the wire, and (ii) the right third of the wire. (d) Without further use of Maple, write down the probability of finding the particle in the left third of the wire. (e) Write down a mathematical expression for the mean value (c) for the ground state of the particle in a wire. Use Maple to evaluate this expres- sion. Check your answer against the text (lecture notes). Hint: Recall the quantum postulate (No. 2) about the relationship between opera- tors and observables, and the postulate (No. 5) about the calculation of average or expectation values. (f) Write down a mathematical expression for the mean value (2) for the ground state of the particle in a wire. Use Maple to evaluate this expression. (g) Use the values of (x) and (**) calculated above to obtain the standard deviation (1). Express (2) as a percentage of the wire's length. Pr 4.4 All quantitative spectroscopy begins with an experimental mea- surement of a wavelength, or frequency, and uses a quantum mechanical model to finish up with a physical property of the system upon which the experiment was made. Use QCEnergies in homework mode, solve one prob- lem per student from the "Translational spectroscopy" topic, and submit the file created by the program to your teaching assistant using the dropbox on the course website. Note that the practice mode of QCEnergies does not create the file needed for you to get a mark.

Answer 1

For maple definite integral. you need to do the following

1. Type the function.
2. Right click a.nd select integrate.
3. You will get an indefinite integral.
4. Then you will see int((I, x);
5. Change int((I, x); to int((I, x=lower limit .. upper limit);
6. Run it again

Pr. 4,3 m at (a) Appendix A is not giren, Henee I will use more general way Canalytical way) for Maple software, just write/type- the function, (2) Sin (tox/a) r For i tyke, type pi thena drop down menu will appear select, on tyking Ctrl+ Space A will hos beo needed to be selected. The particle should be within the wire of length = a . Thus we can say that, the total probability of finding the particle within the length a -1 [The Particle may be at different places but must be found anywhere in the wire So Probability of finding the particle in the wire is. þarty

Therefore the integration limit should be x=0 to x=a. Thus we will have to have for normalitatis (4 4 * dx-1- o Now 4 = (2) ? sin real So its complex conjugate 4* will be equal to y Thus we have a from - 2 Page-2) (4 dx=1-② o Thus for normalization we heed saj sem se da 1 - Let Tx = 0 - 6 i. I dx=do a, da- a do 5 Now when x=0, from 6 Now when x=a, O=K Thus, equation - will be converted to Sa) Sime.com).do = 1-0 0=0 A=T

As pere equation. ③ for normalization we following integrations a sino do I need to calculate the lat sollowina a x 2 is kept inside the integration as asino =1-los 20 Now we are going to calculate the definite integral. Salano do = f(- cos 26) 26 A 0 (do - Coszodo - 07 -Sinze to=0 = (0-0) -(0-0 R. (8 the value from ③ to 6 we have 0-0 Replacing - 2x - x x = 1 ... Thus the wave function R0 - 2220 na is normalized. Ky - Left third [Z12 Middle third Llog Right Third. <-- - a - -

For middle third we need to find the Probability between a=al to x= 2a Hence we need to find (Page 4 20/ 4 + 4 da real dala yada 4 al3 - Taysin 303 4 [243] aj3 x = 20 1962 4T a a = 0.609

) Now, for middle We need to third integrate from x=a 310 22 x=2a x=2a 3 Then we need to find 2app 4².dx a/3 A Ta a 2 T مار . C) For, Right the road city xe = 2a to ze=a wa 2 sim (AZ (Page -asen (2121) 1 7 - ?( dx 4 a da SIM Here the = 0.19550 we can see that the probability of finding particle is higher in the middle third compared to the right third.

d) for Left thard we need to find the probability between x=0 to = a S(2) din Anda 2/3 lasin anda Jal3 1- Cos 200 da 1) page sin ała 21 2- lia - x - Siem 25 202 72=a2 - 24 220 - Sinar -0+0 Sin 2 = sin 2 = - B = 0.19 550 Cano)

Average value of any oberrable, a is given by (1 = ( 4* i v dv î> Operator for the observable all space Heree all space, x=0 to x=a the particle is confined in the wire. 4* = 4 as it is real f = displacement operator dv=da, as it is one dimensional lysten. 1 Pager) y Y as it is real (Page 7 fron equation le scalatin 12a a cay tinta de So equal on 2e becomes (3) siam. xdx - 30 яго М. for (x²7 we need to use x²dx instead of adze 2 un T2 2 a 1 - (3) -

Page-8) = (ta - er at = 0.033.02 (22>-<x72 :.0(x) = = 80,033a2 = 0.181 a Caur