9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) i...
9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) is independent of time, is described by the one-dimensional non-relativistic Schrödinger equation (where m is its mass, h is the reduced Planck constant, i is the imaginary number): 2m (a) Verify that it is a parabolic equation (page E-1-2). [It has wave-like solutions, however.] (b) Use the substitution Px,t)-Xx)Tt) to separate the equation into two ODEs. (c) Solve for T, write its solution into cos and sin terms using Euler's formula (see B-1-23). (d) The separation constant 2 represents the particle's total energy. Based on our classroom observation that eigenvalues are always related by an integer proportion to each other, deduce that the particle's energy levels are discrete in nature, rather than occupying a continuous range. Does this fact agree with your understanding of physics (i.e. the quantum theory)?
9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) is independent of time, is described by the one-dimensional non-relativistic Schrödinger equation (where m is its mass, h is the reduced Planck constant, i is the imaginary number): 2m (a) Verify that it is a parabolic equation (page E-1-2). [It has wave-like solutions, however.] (b) Use the substitution Px,t)-Xx)Tt) to separate the equation into two ODEs. (c) Solve for T, write its solution into cos and sin terms using Euler's formula (see B-1-23). (d) The separation constant 2 represents the particle's total energy. Based on our classroom observation that eigenvalues are always related by an integer proportion to each other, deduce that the particle's energy levels are discrete in nature, rather than occupying a continuous range. Does this fact agree with your understanding of physics (i.e. the quantum theory)?