The ground state wave function for a particle in a one-dimensional box is of length L is y = (2/L)¹² sin(7x/L). Calculate the probability of the particle between x=4.00 nm to x = 4.80 nm. Assume the length of the box is 8.5 nm. Answer Choices: (A) 0.840 (B) 0.143 (C) 0.186 (D) 0.256

P7D.8* A particle is confined to move in a one-dimensional box of length L. If the particle is behaving classically, then it simply bounces back and forth in the box, moving with a constant speed. (a) Explain why the probability density, P(x), for the classical particle is 1/L. (Hint: What is the total probability of finding the particle in the box?) (b) Explain why the average value of x" is (x")= , P(x)x"dx . (c) By evaluating such an integral, find (x) and (x*). (d) For a quantum particle (x)=L/2 and (x*)=L (}-1/2n°n²). Compare these expressions with those you have obtained in (c), recalling that the correspondence principle states that, for very large values of the quantum numbers, the predictions of quantum mechanics approach those of classical mechanics.

Consider a 1D particle in a box confined between a = 0 and x = 3. The Hamiltonian for the particle inside the box is simply given by Ĥ . Consider the following normalized wavefunction 2m dz² ¥(2) = 35 (x³ – 9x). Find the expectation value for the energy of the particle inside the box. Give your 5832 final answer for the expectation value in units of (NOTE: h, not hbar!). In your work, compare the expectation value to the lowest energy state of the 1D particle in a box and comment on how the expectation value you calculated for the wavefunction ¥(x) is an example of the variational principle.

A.1 Answer the following two questions: (i) A particle with spin s = 2 (in units of ħ) has an orbital angular momentum of € = 1. What are the possible values for its total angular momentum J? (ii) A further measurement of J₂ yields J₂ = −2. What are now the possible values of J? ==

please answer #6

(a) What is the lowest possible value of the principal quantum number (n) when the angular momentum quantum number (ℓ) is 1? (b) What are the possible values of the angular momentum quantum number (ℓ) when the principal quantum  number (n) is 4 and the magnetic quantum number (mℓ) is 0?

n = 9.19 (a) Calculate Ax Apx for a particle in a linear box for 1, 2, and 3 using the equation in Example 9.13. Compare these values with the minimum product of uncertainties from the Heisenberg uncertainty principle. (b) What is the uncer- tainty in x for a particle in a 0.2-nm box when its quantum num- ber is unity? In a 2-nm box?

A nitrogen molecule is confined in a cubic box of volume 1.00 m3. (i) Assuming that the molecule has an energy equal to 3/2kT at T = 300 K, what is the value of n = (nx2 + ny2 + nz2)1/2 for this molecule? (ii) What is the energy separation between the levels n and n + 1? (iii) What is the de Broglie wavelength of the molecule?

particle is confined to a one-dimensional box of length L. Deduce the location of the posit ions with in the box at which the particle is most likely to be found when the quantum number of the particle is (a) n = 1. (b) n = 2. and(c) n = 3.