Answered: Find (a) the corresponding Schrödinger…

Find (a) the corresponding Schrödinger equation and wave function, (b) the energy for the infinite-walled well problem of size L, (c) the expected value of x () on the interval [0,a/4], (

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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.
Find the corresponding Schrödinger equation and wave function, The energy for the infinite-walled well problem of size L, The expected value of x (<x>) on the interval [0,a/4], The expected value of p (<p>) for the same interval and the probability of finding at least one particle in the same interval. Do not forget the normalization, nor the conditions at the border.
A particle is confined to a one dimensional box with boundaries at x=0 and x-1. The wave function of the particle within the box boundaries is V(x) 2100 (- x + ) and zero V 619 everywhere else. What is the probability of finding the particle between x=0 and x=0.621? Do not enter your final answer as a percentage, but rather a number between 0 and 1. For instance, if you get that the probability is 20%, enter 0.2.
Consider the potential barrier problem as illustrated in the figure below. Considering the case where E > V0: (a) find the wave function up to a constant (that is, you don't need to compute the normalization constant) (b) Calculate the reflection coefficient of the wave function. This result is expected classically?
Consider a particle moving in a one-dimensional box with walls at x = -L/2 and L/2. (a) Write the wavefunction and probability density for the state n=1. (b) If the particle has a potential barrier at x =0 to x = L/4 (where L = 10 angstroms) with a height of 10.0 eV, what would be the transmission probability of the electrons at the n = 1 state? (c) Compare the energy of the particle at the n= 1 state to the energy of the oscillator at its first excited state.
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 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)
Consider a weakly anharmonic a 1D oscillator with the poten- tial energy m U(x) = w?a² + Ba* 2 Calculate the energy levels in the first order in the small anharmonicity parameter 3 using TIPT and the ladder operators.
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)