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) The image shows a mathematical expression for a potential energy function \( V(x) \). The function is defined piecewise as follows:\[ V(x) = \begin{cases} 0, & 0 < x < L \\\infty, & \text{elsewhere}\end{cases}\]Explanation:- The function \( V(x) \) represents a potential barrier.- It is equal to 0 within the interval \( 0 < x < L \).- Outside of this interval, the potential function \( V(x) \) is infinite (\(\infty\)), indicating an impenetrable barrier. This type of potential is often used in quantum mechanics to describe a particle in an infinite potential well, where the particle is free to move within the interval \( 0 < x < L \) but cannot exist outside of it. The function is defined as follows:\[\Psi(x) = \begin{cases} \left( \frac{1+i}{2} \right) \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) + \frac{1}{\sqrt{2L}} \sin\left(\frac{2\pi x}{L}\right), & 0 < x < L \\0, & \text{elsewhere}\end{cases}\]This describes a piecewise function, where \(\Psi(x)\) takes a specific form involving trigonometric functions and complex numbers for \(0 < x < L\), and is zero elsewhere.

Answered: Image /qna-images/answer/fdeebe55-1ed6-494b-a3c8-0dfa74a83c30.jpg

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 for the particle at time t. (Hint: can be obtained by inspection, without an integral)

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 for the particle at time t. (Hint: can be obtained by inspection, without an integral)