Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction ψn. (a) Without evaluating any integrals, explain why ⟨x⟩ = L/2. (b) Without evaluating any integrals, explain why ⟨px⟩ = 0. (c) Derive an expression for ⟨x2⟩ (the necessary integrals will be found in the Resource section). (d) For a particle in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why <p2x> = n2h2/4L2.

If two wavefunctions, Wa and Wb, are orthonormal and degenerate, then what is true about the linear combinations 1 1 w. +v.) a a and (a) y+ and y- are orthonormal. (b) y+ and y- are no longer eigenfunctions of the Schrödinger equation. (c) V+ and y- have the same energy. (d) V+ and Y- have the same probability density distribution.

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.

A normalized wavefunction for a particle confined between 0 and L in the x direction is ψ = (2/L)1/2 sin(πx/L). Suppose that L = 10.0 nm. Calculate the probability that the particle is (a) between x = 4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm, (c) between x = 9.90 nm and 10.00 nm, (d) between x = 5.00 nm and 10.00 nm.

5. Consider a particle constrained to move in one dimension described by the wavefunction v (x) = Ne2** (a) Determine the normalization constant (b) Is the wavefunction an eigenfunction of d? +16x? dx? (c) Calculate the probability of finding the particle anywhere along the negative x-axis

(a) Look very carefully at the pictures below: Picture 1 and Picture 2. What does each of the pictures represent? Give FULL EXPLANATIONS for your answer. (1) (2) (b) Look very carefully at the picture below. Give the relevant quantum numbers. Explain your answer. у-аxis (c) (i) What is a wavefunction? (ii) What are the two parts of a wavefunction? (iii) What is the importance of squaring a wavefunction? (iv) Where is a wavefunction obtained from?

(a) For a particle in the stationary state n of a one dimensional box of length a, find the probability that the particle is in the region 0xa/4.(b) Calculate this probability for n=1,2, and 3.

4. Given these operators A=d/dx and B=x², can you measure the expectation values of the corresponding observables to infinite precision simultaneously?