= 5.5 The one-dimensional parity operator II is defined by П(x) (-x). In other words, II changes x into -x everywhere in the function. (a) Is II a Hermitian operator? (b) For what potentials V(x) is it possible to find a set of wavefunc- tions which are eigenfunctions of the parity operator and solutions of the one-dimensional time-independent Schrödinger equation?

A good example of time evolution of an operator is the position in x in 1-dimension. This simplified case allows us to deal with the wave function as just a scalar so, = ƒ ¥¹ â¥dx In classical physics, we define momentum as pa m dx dt - -1 8² 2m əx² Using the spatial representation in 1D, momentum is p = -2 -ithe free particle Hamiltonian is, H Calculate the time derivative of and use the Schrodinger's Equation id/dt = Hy for a free particle to derive the quantum momentum as follow, = d dt mvx = m.

goes from - to +∞. 18. Normalize the wavefunction, = (2-)e and find and . 2018

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A particle of mass m moves in a three-dimensional box of sides a, b, c. If the potential is zero inside and infinity outside the box, find the energy eigenvalues and eigenfunctions.

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Legrende polynomials The amplitude of a stray wave is defined by: SO) =x (21+ 1) exp li8,] sen 8, P(cos 8). INO Here e is the scattering angle, / is the angular momentum and 6, is the phase shift produced by the central potential that performs the scattering. The total cross section is: Show that: 'É4+ 1)sen² 8, .

how do i do a? I sit by finding the constant = 1? Pls help me

P7E.11 The expectation value of the kinetic energy of a harmonic oscillator is most easily found by using the virial theorem, but in this Problem you will find it directly by evaluating the expectation value of the kinetic energy operator with the aid of the properties of the Hermite polynomials given in Table 7E.1. (a) Write the kinetic energy operator T in terms of x and show that it can be rewritten in terms of the variable y (introduced in eqn 7E.7) and the frequency w as d? dy The expectation value of this operator for an harmonic oscillator wavefunction with quantum number v is d? (T),=-hwaN; ſ H,en. dy H,evirdy where N, is the normalization constant (eqn 7E.10) and a is defined in eqn 7E.7 (the term a arises from dx = ady). (b) Evaluate the second derivative and then use the property H" – 2yH, + 2vH, = 0, where the prime indicates a derivative, to rewrite the derivatives in terms of the H, (you should be able to eliminate all the derivatives). (c) Now proceed as in the text, in…

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