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…

b. Prove that the conserved charge in energy-momentum tensor is given by TO p

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

Q5: Consider a particle of mass m in a two-dimensional box having side length L and L₁ with L = 2L, and V=0 in the box, ∞ outside; Suppose V=10 J in the box. What effect has this on the eigenvalues? the eigen functions?

Normalize the following wavefunction and solve for the coefficient A. Assume that the quantum particle is in free-space, meaning that it is free to move from x € [-, ∞]. Show all work. a. Assume: the particle is free to move from x € [-0, 00] b. Wavefunction: 4(x) = A/Bxe¬ßx²

Consider the following wave function. TT X a = B sin(- (x) = E 2 π.χ. a −) + C · sin(² a. Does this function describes a particle-in-a-box acceptable wave function? Name the conditions to be fulfilled. b. Is this function an eigenfunction of the total energy operator H when H is the Hamilton operator.

A particle is placed in the potential well of finite depth U The width a of the well is fixed in such a way that the particle has only one bound state with binding energy e Uu/2 Calculate the probabilities of finding the particle in classically allowed and classically forbidden Tegiens. 2./10/Calculate the result of the transformation of the voctor operator projection R, by rotation R, around an angle a. Hint Take the second derivative of the transformed operator with respect to a and solve the second-order "uoenbo