4.7 a. Let y(x.t) be the wave function of a spinless particle corresponding to a plane wave in three dimensions. Show that y* (x.-) is the wave function for the plane wave with the momentum direction reversed. b. Let x(n) be the two-component eigenspinor of a-n with eigenvalue +1. Using the explicit form of x() (in terms of the polar and azimuthal angles ß and y that characterize f) verify that -io₂x*() is the two-component eigenspinor with the spin direction reversed.

40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow the

please answer c) only 2. a) A spinless particle, mass m, is confined to a two-dimensional box of length L. The stationary Schrödinger equation is - +a) v(x, y) = Ev(x, y), for 0 < r, y < L. The bound- ary conditions on ý are that it vanishes at the edges of the box. Verify that solutions are given by 2 v(1, y) sin L where n., ny = 1,2..., and find the corresponding energy. Let L and m be such that h'n?/(2mL²) = 1 eV. How many states of the system have energies between 9 eV and 24 eV? b) We now consider a macroscopic box (L of order cm) so that h'n?/(2mL?) ~ 10-20 eV. If we define the wave vector k as ("", ""), show that the density of states g(k), defined such that the number of states with |k| between k and k +dk is given by g(k)dk, is Ak 9(k) = 27 c) Use the expression for g(k) to show that at room temperature the partition function for the translational energy of a particle in a macroscopic 2-dimensional box is Z1 = Aoq, where 2/3 oq = ng = mk„T/2nh?. Hence show that the average…

Consider a particle of mass m which can move freely along the x axis anywhere from x=-a/2 to x=a/2, but which is strictly prohibited from being found outside this region (technically this is the infinite square well potential). 1. Verify that the attached wavefunction is a solution of the Schrodinger's equation. Consider V=0.  2. Deduce the value of the energy E. 3. Verify the  uncertainty principle