Particle in a potential well

A particle of mass m moves in the potential V(x) = -g*delta(x) x>-a infinity x<-a (delta (x) = dirac delta function) a. Without worrying about continuity or boundary conditions, write down the general solution of the Schrodinger equation for a bound state (energy E<0) in regions( -a less than x less than 0) and x>0. b. In terms of your solution in part (a), write down the boundary conditions at x=-a and continuity conditions at x=0. c. From the conditions in part (b), derive a transcendental equation determining the bound state energy E. Solve the equation in the limit as a approaches infinity.

Time dependent Schrodinger equation (TDSE) in operator from

Time dependent Schrodinger equation (TDSE) in operator from. (Pls refer to the attached jpg)

Estimate the ground state energy of a particle in the potential V(x) = lambda *(x)^4 using variational methods and the uncertainty principle

Estimate the ground state energy of a particle of mass m moving in the potential V(x) = lambda *(x)^4 by two different methods. a. Using the Heisenberg Uncertainty Principle; b. Using the trial function psi(x)=N*e^{[- abs(x)]/(2a)} where a is determined by minimizing (E) *Note abs = absolute value

Find Hermitian Conjugate of (d/dx)(x^2)(d^2/dx^2) + (1 + 3i)xd/dx

Note that all the derivatives in the problem are partials with respect to x, and the i is an imaginary number Find the Hermitian Conjugate (operator O-dagger) of the operator O (operator) = (d/dx)*(x^2)*(d^2/dx^2) + (1 + 3i)*x*d/dx

1D motion of a quantum particle in an external potential with it's total energy

1D motion of a quantum particle in an external potential with it's total energy given by: (Please refer to the attached jpg)