2. According to Quantum Mechanics, the energy levels of a freeparticle is quantized. It can only take specific numbers not justany values. This is one of the fundamental difference fromClassical Mechanics. This quantization can be shown in a simpleparticle in a box approximation. According to that a free particlein a 1-D box can be shown as:The particle experiences no potential energy inside the box. Thewalls of the box are infinitely large so that the particle cannotescape from it.a- According to the second boundary condition of y(a)=0, whatare the possible values of B?b- According to normalization rule, the probability of findingthe particle from 0 to a must be 1. That is;Using this rule and updated wave function find the value ofsin zdz2)%3DB? (Hint: Joc- According to the value of B, update wave function. Now youhave solved (time independent) Schrodinger equation forparticle in a box case!d- Using the non-zero values of B, update the value of E interms of a, m, h and n=1,2,... Now you have found all theenergy levels in a particle in 1-D box, which corresponds tothe energy levels of Hydrogen atom.

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7. One electron is trapped in a one-dimensional square well potential with infinitely high sides. a. If you have a probe that has a width for electron detection Ax = 0.00350L in the x direction, for the first excited state ( n =2), what is the probability that the electron is found in the probe when it is centered at x = L/4, (hint: you can use an approximation for this - you do not need to do an integral)? b. What is the average number of electrons that you would detect using the probe described in part "b." centered at x = L/4, ifthe electron is in the first excited state (n = 2) for each experiment and you repeat the experiment N, =100,000 times?
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