A particle in a one dimensional infinite square well (box) is in a superposition of two quantum states:1 21234 where1 2and are orthogonal, normalized (ortho-normal) eigenfunctions of the Hamiltonian.(a) Calculate the new normalization constants for1 2and . [2 marks](b) A measurement of energy is made. What is the probability that the measurement will find energy1E ?What is the probability that the measurement will find energy2E ? [2 marks](c) Write down the integral expression for the evaluation of the expectation value of position,x - evaluateany “trivial” integrals, but leave as integrals any that require a full solution. [2 marks](d) Write down the integral expression for the evaluation of the expectation value of momentum,xp -evaluate any “trivial” integrals, but leave as integrals any that require a full solution
Question
A particle in a one dimensional infinite square well (box) is in a superposition of two quantum states:1 21234 where1 2and are orthogonal, normalized (ortho-normal) eigenfunctions of the Hamiltonian.(a) Calculate the new normalization constants for1 2and . 2 marks A measurement of energy is made. What is the probability that the measurement will find energy1E ?What is the probability that the measurement will find energy2E ? 2 marks Write down the integral expression for the evaluation of the expectation value of position,x - evaluateany “trivial” integrals, but leave as integrals any that require a full solution. 2 marks Write down the integral expression for the evaluation of the expectation value of momentum,xp -evaluate any “trivial” integrals, but leave as integrals any that require a full solution
Solution
(a) The wave function of a quantum system must be normalized. This means that the total probability of finding the particle somewhere in space must be 1. Given that ψ1 and ψ2 are orthogonal and normalized, their normalization constants are both 1.
(b) The probability of finding the particle in a particular energy state is given by the square of the coefficient of that state in the wave function. Therefore, the probability of finding the particle with energy E1 is |1|^2 = 1 and the probability of finding the particle with energy E2 is |2|^2 = 4.
(c) The expectation value of position, <x>, in quantum mechanics is given by the integral ∫ψ* x ψ dx, where ψ* is the complex conjugate of the wave function, x is the position operator, and the integral is taken over all space.
(d) The expectation value of momentum, <p>, is given by the integral ∫ψ* (-iħ d/dx) ψ dx, where -iħ d/dx is the momentum operator. The integral is taken over all space.
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