solve part a and b The first two normalized wave functions for a harmonic oscillator are as follows: (α is a constant, $-\infty \leq x \leq +\infty$)**Ground state:**\[\psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/4} e^{-\alpha x^2/2}\]**First excited state:**\[\psi_1(x) = \left( \frac{4\alpha^3}{\pi} \right)^{1/4} x e^{-\alpha x^2/2}\](a) Verify that the above two wave functions are orthonormal.**Standard integral:**\[\int_{-\infty}^{+\infty} e^{-\alpha x^2} \, dx = \left( \frac{\pi}{\alpha} \right)^{1/2}\]\[\int_{-\infty}^{+\infty} x^2 e^{-\alpha x^2} \, dx = \frac{\sqrt{\pi}}{2\alpha^{3/2}} = \left( \frac{\pi}{4\alpha^3} \right)^{1/2}\](b) What is the average displacement $\langle x \rangle$ of the harmonic oscillator in the ground state?

Given Information:The normalized wave function is given as:Ground state: .First Excited state: .Here, is the constant and .To find:a) The two functions are orthonormal.b) Average displacement in ground state.Concept used:If the two vector functions and are orthogonal, then .A vector functions and are orthonormal only if they are orthogonal to each other and and , where is the complex conjugate of respectively.A function is said to be an odd function only if .For an odd function , .a)First, show the functions on ground state and excited state is orthogonal to each other.Find the product of two functions.Now, check it is of odd or even function property.This follows the property of odd function.Hence, it is of odd function.By the property of intergral on odd function,.Therefore, the wave functions are orthogonal to each other.As the wave functions are real functions, they has no complex part.Therefore, their complex conjugate is also same as the function.i.e., .Now, find the value of .Now, find the value of .As the functions satisfy the condition to be orthonormal, the two wave functions are orthonormal.b)Average displacement of function at ground state can be determined as .Now, it is required to find the displacement of at the ground state.The required average displacement is 0 at the ground state.a) The provided two wave functions are orthonormal.It is verified with the use of basic definition and the given standard integral values.b) The average displacement of at the ground state is 0.