For an electron in a one-dimensional box of width L (x lies between 0 and L), (a)Write down its wavefunction and the allowed energy. (b)If the electron is in a superposition of the ground state and second excited state, write down the wavefunction and compute the probability of finding the electron at 1/6 ?. (Don’t forget to normalize it!)

1) a) A particle is in an infinite square well, with ground state energy E1. The wavefunction is 3 *y. Find in terms of E1. (There is an easy way to do this; no actual integrals 4 + 5 required.) b) A particle is in an infinite square well, with ground state energy Ej. Find a normalized wavefunction that has a total energy expectation value equal to 3E1. (It will be a superposition.) Keep all your coefficients real and positive. c) Now time-evolve your answer from part b, to show how the wavefunction varies with time.

Consider an electron in a one-dimensional, infinitely-deep, square potential well of width d. The electron is in the ground state. (a) Sketch the wavefunction for the electron. Clearly indicate the position of the walls of the potential well on your sketch. (b) Briefly explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.

The normalised wavefunction for an electron in an infinite 1D potential well of length 80 pm can be written:ψ=(0.587 ψ2)+(0.277 i ψ7)+(g ψ6). As the individual wavefunctions are orthonormal, use your knowledge to work out |g|, and hence find the expectation value for the energy of the particle, in eV.

The following Eigen function is a typical solution of the time-independent Schrödinger equation and satisfies boundary conditions for a particle in a confined space of a certain length. y(x) = sin (~77) (a) Plot the wave function as a function of x for L = 30 cm and n = 1, 2, 3 and 4. Note: You will need to have 4 plots in the same graph. (b) On a separate graph, plot the probability density (112) as a function of x using the conditions specified in part (a). Note: You will need to have 4 plots in the same graph. (c) Report your observations for parts (a) and (b)

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At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image.  Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral)