Question

Midterm Exam Physical Electronics 1. Consider 1-dimensional infinite square-well potential shown in the figure below: V(x) +? 0 +? x L V(x) = 0, 0?x?L V(x) = +?, otherwise (a) What is the time-independent Schrödinger's equation within the region 0 < x < L for a particle with mass 'm'?(5 pts) (b) Show that the nth energy eigenfunction ($\psi_n$) and the energy eigenvalue ($E_n$) of the particle in the infinite square well are given by: (10 pts) $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi}{L}x)$, $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where n = 1, 2, 3, 4, .... (c) Express the general form of wavefunction $\Psi(x, t)$ with $\psi_n$, $E_n$, and $\Psi(x, 0)$. Here 't' denotes the time. (5 pts)

          Midterm Exam Physical Electronics
1. Consider 1-dimensional infinite square-well potential shown in the figure below:
V(x)
+?
0
+?
x
L
V(x) = 0, 0?x?L
V(x) = +?, otherwise
(a) What is the time-independent Schrödinger's equation within the region 0 < x < L for a
particle with mass 'm'?(5 pts)
(b) Show that the nth energy eigenfunction ($\psi_n$) and the energy eigenvalue ($E_n$) of the particle in
the infinite square well are given by: (10 pts)
$\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi}{L}x)$, 
$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where n = 1, 2, 3, 4, ....
(c) Express the general form of wavefunction $\Psi(x, t)$ with $\psi_n$, $E_n$, and $\Psi(x, 0)$. Here 't' denotes
the time. (5 pts)

Show more…

Midterm Exam Physical Electronics
1. Consider 1-dimensional infinite square-well potential shown in the figure below:
V(x)
+?
0
+?
x
L
V(x) = 0, 0?x?L
V(x) = +?, otherwise
(a) What is the time-independent Schrödinger's equation within the region 0 < x < L for a
particle with mass 'm'?(5 pts)
(b) Show that the nth energy eigenfunction () and the energy eigenvalue (En) of the particle in
the infinite square well are given by: (10 pts)
(x) = √((2)/(L))sin((nπ)/(L)x),
En = (n^2π^2ħ^2)/(2mL^2), where n = 1, 2, 3, 4, ....
(c) Express the general form of wavefunction Ψ(x, t) with , En, and Ψ(x, 0). Here 't' denotes
the time. (5 pts)

Added by Shawn M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Timothy James

12/26/2022

Step 1

Therefore, the time-independent Schrödinger equation becomes: $-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} = E\psi(x)$ Show more…

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Midterm Exam Physical Electronics 1. Consider 1-dimensional infinite square-well potential shown in the figure below: V(x) +∞ 0 +∞ x L V(x) = 0, 0 ≤ x ≤ L V(x) = +∞, otherwise (a) What is the time-independent Schrödinger's equation within the region 0 < x < L for a particle with mass 'm'?(5 pts) (b) Show that the nth energy eigenfunction ($\psi_n$) and the energy eigenvalue ($E_n$) of the particle in the infinite square well are given by: (10 pts) $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi}{L}x)$, $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where n = 1, 2, 3, 4, .... (c) Express the general form of wavefunction $\Psi(x, t)$ with $\psi_n$, $E_n$, and $\Psi(x, 0)$. Here 't' denotes the time. (5 pts)

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00:01 All right, so let's say we have a one -dimensional infinite potential well where this level is infinity and this width is l.

00:09 This is zero.

00:10 This is l.

00:12 You want to write the time independent shortening equation inside the well.

00:18 So basically what we'll just have is negative h squared over 2m times d squared, si, d x squared, equals e times si, because we have no potential.

00:30 Potential in this region.

00:34 And for part b, we want to find the normalization or the normalized wave function given these expressions.

00:42 So what we need, we need si at l to equal psi at zero, which needs to be zero.

00:53 And so si of zero being zero means that b, our coefficient b has to be zero.

01:00 And so si of has to be some constant a times the sign of we'll just write it k -n times x where k -n is equal to n pi over l and then our wave function a our normalization constant a is basically going to equal two over the square root of l so then we want to find the expectation values of x and p and p squared so the expectation value of x is actually kind of simple because it's it's the integral from 0 to l of x times the sine squared of kx dx.

01:42 Sign squared is an even function, x.

01:47 Well, let's just write it this way.

01:49 And if we solve for this, this should just be l over 2 for the first excited state...

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