Consider a particle of mass u bound in an infinite square potential energy well in three dimensions: U(x, y, z) = {+00 0
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- Compute lz for a RR in the l = 1, ml = 1 state using the appropriate wavefunction and operator. Is this the eigenvalue what you expected? Briefly explainConsider a particle in a one-dimensional rigid box of length a. Recall that a rigid box has U (x) = 00 for x a, and U () = 0 for 0 )Consider a particle of mass m moving in the following potential V(x)= v1 for xv2. What is the grown state energy? And the normalized ground sate funcion
- 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?For the potential-energy functions shown below, spanning width L, can a particle on the left side of the potential well (x 0.55L)? If not, why not? b. c. 0 a. 0o 00 E E E-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 "uoenboO Consider the kinetic energy matrix elements between Hydrogen states (n' = 4, l', m'| |P|²| m -|n = 3, l, m), = for all the allowed l', m', l, m values. What kind of operator is the the kinetic energy (scalar or vector)? Use this to determine the following. For what choices of the four quantum numbers (l', m', l, m) can the matrix elements be nonzero (e.g. (l', m', l, m) (0, 0, 0, 0),...)? Which of these nonzero values can be related to each other (i.e. if you knew one of them, you could predict the other)? In this sense, how many independent nonzero matrix elements are there? (Note: there is no need to calculate any of these matrix elements.)
- We can use a quartic function function to represent this potential as shown below. Using the first order perturbation theory for particle in a box, calculate the ground- state energy: V(2) = ca 0< x < b a. How large of an effect on the energy is the perturbation of a curved wall?I need the answer as soon as possibleB3