Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of...
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![Page 1: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/1.jpg)
Lecture 9Particle in a rectangular well
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
![Page 2: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/2.jpg)
Motion in two ormore dimensions
The particle in a rectangular well extends the previous 1D problem to 2D. This introduces two important concepts:
Separation of variables – a very powerful and general technique in reducing the dimension of differential equations.
Degenerate eigenfunctions.
![Page 3: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/3.jpg)
The particle in a rectangular well
The Schrödinger equation for this is:
Boundary conditions are:
1
2
( , ) 0, 0,
0,
x y x L x
y L y
![Page 4: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/4.jpg)
Separation of variables
When a differential equation is two or higher dimensional such as
We must always attempt separation of variables. With this, a 2D problem breaks down into two 1D problems. This happens if the solution is the product of functions of each of the variables Ψ = X(x)Y(y) with no cross term like Z(x,y).
![Page 5: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/5.jpg)
Separation of variables
To see separation of variables indeed occurs, we first assume it does and write the solution in the product form:
![Page 6: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/6.jpg)
Separation of variables The partial derivative ∂2/∂x2 will act only on the
X(x) part (similarly for ∂2/∂y2 on Y), hence
Divide by XY the both sides.
It has the form: f(x) + g(y) = e. This immediately means f(x) and g(y) are individually constant. Separation of variable indeed took place.
![Page 7: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/7.jpg)
F(x) + G(y) = constant
![Page 8: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/8.jpg)
Separation of variables
These are the particle in a box equations!
![Page 9: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/9.jpg)
Separation of variables
1
1/ 2
1
1 1
2( ) sinn
n xX x
L L
2
1/ 2
2
2 2
2( ) sinn
n yY y
L L
1 2
1 21/ 2
1 21 2
2( , ) ( ) ( ) sin sinn n
n x n yx y X x Y y
L LL L
1
2 21
218n
n hE
mL
2
2 22
228n
n hE
mL
1 2 1 2
2 2 2 21 2
, 2 21 28 8n n n n
n h n hE E E
mL mL
1 20 ;0x L y L
![Page 10: Lecture 9 Particle in a rectangular well (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e0d5503460f94af6ab2/html5/thumbnails/10.jpg)
The particle in a three-dimensional box
The argument can be easily extended to 3D:
We now have three quantum numbers.
1/ 2
31 2
1 2 3 1 2 3
8( , , ) sin sin sin
n zn x n yx y z
L L L L L L
1 2 3 1 2 3
2 22 2 2 231 2
, , 2 2 21 2 38 8 8n n n n n n
n hn h n hE E E E
mL mL mL
1 2 30 ;0 ;0x L y L z L
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Degeneracy
Let us suppose L1 = L2 = L in the 2D case. Then the energy is,
This expression gives identical energy for (n1, n2) = (2,1) or (1,2). We say the energy is doubly degenerate in that two different eigenfunctions correspond to this eigenvalue.
1 2
2 2 2 2 22 21 2
, 1 22 2 2( )
8 8 8n n
n h n h hE n n
mL mL mL
2
2
5
8
h
mL
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Degeneracy
The degeneracy is often caused by high symmetry. For the square well case, (n1, n2) = (2,1) and (1,2) wave functions are related by 90° rotation around the center.
These two states are distinguished by different probability densities.
90° rotation
90° rotation
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Degeneracy
When two wave functions Ψ1 and Ψ2 are doubly degenerate:
Then any linear combination of Ψ1 and Ψ2 is also an eigenfunction with the same eigenvalue.
We can use this property to make Ψ1 and Ψ2 orthogonal to each other.
1 1 2 2ˆ ˆ and H E H E
1 1 2 2 1 1 2 2H c c E c c
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Summary
We have introduced the powerful separation of variables technique for differential equations in two or higher dimensions. All two- and higher-dimensional Schrödinger equations we study in this course depends on this powerful technique.
Some eigenvalues are degenerate – more than one eigenfunctions correspond to one eigenvalue.