Question

The time-dependent Schrödinger equation for a free particle in one dimension is [ i hbar frac{partial psi(x, t)}{partial t}=-frac{hbar^{2}}{2 m} frac{partial^{2} psi(x, t)}{partial x^{2}}, ] which can be used to describe the behavior of an electron confined into one dimension (like a very thin wire). In the equation ( hbar ) is Planck's constant ( h ) divided by ( 2 pi, m ) is the mass of the particle, ( i ) is the imaginary unit ( sqrt{-1} ), and ( psi(x, t) ) is the particle's wave function. (a) (10 points) Apply the separation of variables method to come up with two ordinary differential equations, one for each of the two independent variables ( x ) and ( t ), that are related to each other through the chosen linking constant. Note that the time derivative is just first order. Please do not change this. Your solution to this part should explicitly show the steps for separating the variables and justification of your choice of linking constant (see part b). (b) (10 points) For this part, write out the solution to the differential equation for the independent variable ( x ). Since we want the particle confined to a one-dimensional box of width ( L ) with impenetrable walls, impose the periodic boundary conditions ( psi(x=0, t)=0 ) and ( psi(x=L, t)=0 ). What ( X(x) ) results from applying these boundary conditions and what is the value of the linking constant? (c) (5 points) For this part, solve the time-dependent differential equation. There is no condition to apply here. (d) (5 points) Combine the solutions of the two ordinary differential equations to specify ( psi(x, t) ). There should only be one undetermined coefficient at this time. This expression is the energy eigenfunction for this system. In the interest of time, I end the problem here, but we could continue and assume some values for the wire and determine the energy corresponding to the ( mathrm{n}^{ ext {th }} ) eigenfunction. You'll learn more about this in PHY 314.

          The time-dependent Schrödinger equation for a free particle in one dimension is
[
i hbar frac{partial psi(x, t)}{partial t}=-frac{hbar^{2}}{2 m} frac{partial^{2} psi(x, t)}{partial x^{2}},
]
which can be used to describe the behavior of an electron confined into one dimension (like a very thin wire). In the equation ( hbar ) is Planck's constant ( h ) divided by ( 2 pi, m ) is the mass of the particle, ( i ) is the imaginary unit ( sqrt{-1} ), and ( psi(x, t) ) is the particle's wave function.
(a) (10 points) Apply the separation of variables method to come up with two ordinary differential equations, one for each of the two independent variables ( x ) and ( t ), that are related to each other through the chosen linking constant. Note that the time derivative is just first order. Please do not change this. Your solution to this part should explicitly show the steps for separating the variables and justification of your choice of linking constant (see part b).
(b) (10 points) For this part, write out the solution to the differential equation for the independent variable ( x ). Since we want the particle confined to a one-dimensional box of width ( L ) with impenetrable walls, impose the periodic boundary conditions ( psi(x=0, t)=0 ) and ( psi(x=L, t)=0 ). What ( X(x) ) results from applying these boundary conditions and what is the value of the linking constant?
(c) (5 points) For this part, solve the time-dependent differential equation. There is no condition to apply here.
(d) (5 points) Combine the solutions of the two ordinary differential equations to specify ( psi(x, t) ). There should only be one undetermined coefficient at this time. This expression is the energy eigenfunction for this system.

In the interest of time, I end the problem here, but we could continue and assume some values for the wire and determine the energy corresponding to the ( mathrm{n}^{	ext {th }} ) eigenfunction. You'll learn more about this in PHY 314.

$The time-dependent Schrödinger equation for a free particle in one dimension is [ i hbar fracpartial psi(x, t)partial t=-frachbar^22 m fracpartial^2 psi(x, t)partial x^2, ] which can be used to describe the behavior of an electron confined into one dimension (like a very thin wire). In the equation ( hbar ) is Planck's constant ( h ) divided by ( 2 pi, m ) is the mass of the particle, ( i ) is the imaginary unit ( sqrt-1 ), and ( psi(x, t) ) is the particle's wave function. (a) (10 points) Apply the separation of variables method to come up with two ordinary differential equations, one for each of the two independent variables ( x ) and ( t ), that are related to each other through the chosen linking constant. Note that the time derivative is just first order. Please do not change this. Your solution to this part should explicitly show the steps for separating the variables and justification of your choice of linking constant (see part b). (b) (10 points) For this part, write out the solution to the differential equation for the independent variable ( x ). Since we want the particle confined to a one-dimensional box of width ( L ) with impenetrable walls, impose the periodic boundary conditions ( psi(x=0, t)=0 ) and ( psi(x=L, t)=0 ). What ( X(x) ) results from applying these boundary conditions and what is the value of the linking constant? (c) (5 points) For this part, solve the time-dependent differential equation. There is no condition to apply here. (d) (5 points) Combine the solutions of the two ordinary differential equations to specify ( psi(x, t) ). There should only be one undetermined coefficient at this time. This expression is the energy eigenfunction for this system. In the interest of time, I end the problem here, but we could continue and assume some values for the wire and determine the energy corresponding to the ( mathrmn^ ext th ) eigenfunction. You'll learn more about this in PHY 314.$

Added by Stephen C.

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