今回は三次元のシュレディンガー方程式を変数分離で解いていきたいと思います。
三次元といっても、一次元のときとやることは変わらないので落ち着いて解いていきましょう。
問題設定
質量\(m\)の粒子が箱の中で運動している.箱の3辺の長さは,\(x,y,z\)軸方向でそれぞれ\(a,b,c\)である。また, 箱の中のポテンシャル\(V\)はゼロとし,箱の外部では\(V= \infty\)とする。波動関数は連続で, 箱の外ではゼロであり, 箱の中で規格化されているものとする.
解法
変数分離
この粒子の時間に依存しないシュレーディンガー方程式 $$ \left(-\frac{\hbar^2}{2m}\nabla^2\right)\Psi = E\Psi \tag{1} $$
を解いて求めます。(\(\Psi(x,y,z)\)は波動関数,\(E\)はエネルギー固有値,箱の中を考えるのでV = 0)
まず、 $$ \Psi = \psi_x(x)\psi_y(y)\psi_z(z)\tag{2} $$ と変数分離を行います。
(1)式に(2)式を代入すると、 $$ -\frac{\hbar^2}{2m}\left(\frac{\partial^2 \psi_x(x)}{\partial x^2}\psi_y(y)\psi_z(z) + \psi_x(x)\frac{\partial^2 \psi_y(y)}{\partial y^2}\psi_z(z) + \psi_x(x)\psi_y(y)\frac{\partial^2 \psi_z(z)}{\partial z^2}\right)\\ = E\psi_x(x)\psi_y(y)\psi_z(z) $$ なので $$ -\frac{\hbar^2}{2m}\left(\frac{1}{\psi_x(x)}\frac{\partial^2 \psi_x(x)}{\partial x^2}+ \frac{1}{\psi_y(y)}\frac{\partial^2 \psi_y(y)}{\partial y^2}+ \frac{1}{\psi_z(z)}\frac{\partial^2 \psi_z(z)}{\partial z^2}\right) = E\tag{3} $$ となります。
ここで、\(\frac{1}{\psi_x(x)}\frac{\partial^2 \psi_x(x)}{\partial x^2}\)に注目して
(3)式を $$ -\frac{\hbar^2}{2m}\left(\frac{1}{\psi_x(x)}\frac{\partial^2 \psi_x(x)}{\partial x^2}\right) = \frac{\hbar^2}{2m}\left(\frac{1}{\psi_y(y)}\frac{\partial^2 \psi_y(y)}{\partial y^2} + \frac{1}{\psi_z(z)}\frac{\partial^2 \psi_z(z)}{\partial z^2}\right) + E \tag{4} $$ と変形します。
ここで(4)式の左辺に注目すると、\(x\)のみの関数、右辺に注目すると\(y,\;z\)のみの関数なので、
$$ -\frac{\hbar^2}{2m}\left(\frac{1}{\psi_x(x)}\frac{\partial^2 \psi_x(x)}{\partial x^2}\right) = e_x(定数) $$ と置くことができます。
これを変形すると、 $$ \frac{\partial^2 \psi_x(x)}{\partial x^2} = -\frac{2me_x}{\hbar^2}\psi_x(x) $$ となり、これを解くと、
$$ \psi_x(x) = A\cos kx + B\sin kx\;\;\;\left(ただし、k = \sqrt{\frac{2me_x}{\hbar^2}},\;A,Bは定数\right) \tag{5} $$ となります。
境界条件
また、境界条件として
$$ \begin{eqnarray} \begin{cases} \psi_x(0) = 0 \\ \psi_x(a) = 0 \end{cases} \end{eqnarray} $$ を考えると、(5)式から、
$$ \begin{eqnarray} \begin{cases} \psi_x(0) = A = 0\\ \psi_x(a) = A\cos ka + B\sin ka = 0 \end{cases} \end{eqnarray} $$ より、 $$ B\sin ka = 0 $$ が求まります。
これを満たす\(ka\)は $$ ka = l\pi\;\;\;(l = 1,2,3,\cdots) $$ なので、
$$ k = \frac{l\pi}{a} \tag{6} $$
となります。
固有エネルギー
このことから、固有エネルギー\(e_x\)を求めることができます。
(5)式で\(k = \sqrt{\frac{2me_x}{\hbar^2}}\)だったので, $$ k = \frac{l\pi}{a} = \sqrt{\frac{2me_x}{\hbar^2}} $$ となります。中辺と右辺を変形すると、
$$ e_x = \frac{\hbar^2l^2\pi^2}{2ma^2}\tag{7} $$ となり、エネルギー固有値を求めることができました。
固有関数
以上から\(\psi_x(x)\)は
$$ \psi_x(x) = B\sin \left(\frac{l\pi}{a}\right)x $$ となりました。
\(\psi_y(y),\;\psi_z(z)\)も同様にして
$$ \psi_y(y) = C\sin \left(\frac{m\pi}{b}\right)y \\ \psi_z(z) = D\sin \left(\frac{n\pi}{c}\right)z \\(ただし、n,m = 1,2,3,\cdots ,\;C,Dは定数) $$ となります。
よって元の関数は $$ \Psi = C_0\sin \left(\frac{l\pi}{a}x\right)\sin \left(\frac{m\pi}{b}y\right)\sin \left(\frac{n\pi}{c}z\right)\\ (ただし、C_0 = BCD) $$ となります。
規格化
ここで、\(\Psi\)を規格化し、\(C_0\)を求めてみましょう。
全領域にわたって積分しますが、箱の中しか波動関数が存在できないので箱の中の領域だけで積分します。
$$ \int_{0}^{a}\int_{0}^{b}\int_{0}^{c} dxdydz\;\;|\Psi|^2 = 1 $$ なので $$ \int_{0}^{a}\int_{0}^{b}\int_{0}^{c} dxdydz\;\;C_0^2\sin^2 \left(\frac{l\pi}{a}x\right)\sin^2 \left(\frac{m\pi}{b}y\right)\sin^2 \left(\frac{n\pi}{c}z\right) = 1 $$ となります。
ここで、 $$ \int_{0}^{a}dx\;\sin^2 \left(\frac{l\pi}{a}x\right) = \int_{0}^{a}dx\;\frac{1-\cos\left(2\frac{l\pi}{a}x\right)}{2} = \frac{a}{2} $$ より、
$$ \begin{eqnarray} \int_{0}^{a}\int_{0}^{b}\int_{0}^{c} dxdydz\;\;C_0^2\sin^2 \left(\frac{l\pi}{a}x\right)\sin^2 \left(\frac{m\pi}{b}y\right)\sin^2 \left(\frac{n\pi}{c}z\right) &=& C_0^2\frac{a}{2}\frac{b}{2}\frac{c}{2}\\ &=& C_0^2\frac{abc}{8}\\ &=& 1 \end{eqnarray} $$ より、
$$ C_0 = \sqrt{\frac{8}{abc}} $$ となります。
よって、 $$ \Psi = \sqrt{\frac{8}{abc}}\sin \left(\frac{l\pi}{a}x\right)\sin \left(\frac{m\pi}{b}y\right)\sin \left(\frac{n\pi}{c}z\right) $$ となります。
固有エネルギー（全体）
最後に固有エネルギーを求めましょう。
これは各成分の固有エネルギーの和になるので
$$ E = e_x + e_y + e_z $$ なので(7)式で求めたように\(e_y,e_z\)ももとめると、 $$ e_y = \frac{\hbar^2n^2\pi^2}{2mb^2}\\ e_z = \frac{\hbar^2m^2\pi^2}{2mc^2} $$ となります。
よって $$ E = \frac{\hbar^2\pi^2}{2m}\left(\frac{l^2}{a^2} + \frac{n^2}{b^2} + \frac{m^2}{c^2}\right) $$ となります。
参考
コメント
I need to to thank you for this excellent read!! I absolutely enjoyed every bit
of it. I have you book-marked to lolok at new stuff you post…
whoah this blog is great i love studying your posts.
Keep up the good work! You understand, many individuals are searching round for this information, you can help them
greatly.
Hi there excellent website! Does running a blog suchh as this
require a large amount of work? I’ve no knowledge oof computer programming however I was hoping to
start my owwn blog in the near future. Anyhow, should you have any
suggestions or techniques for new blog owners please share.
I know this is off subject buut I simply wanted to ask. Appreciate it!
I wanted to thank you for this good read!! I certainly loved every bit
of it. I have got you book marked to check out new stuff you post…
I have read so many posts on the topic of the blogger
lovers except this article is genuinely a nice paragraph, keep it up.
Have you ever considered publishing an e-book or guest authoring on other blogs?
I have a blog based on the same topics you discuss and would really like to have you share some stories/information. I
know my visitors would enjoy your work. If you are even remotely interested,
feel free to send me an email.
Every weekend i used to pay a visit this web site, for the reason that i wish for enjoyment, as this this
web site conations really pleasant funny information too.
If you are going for most excellent contents like I do,
simply pay a quick visit this web site daily
as it offers quality contents, thanks
It’s going to be end of mine day, however before end I am reading this enormous post
to increase my knowledge.
I am not sure where you’re getting your information, but great topic.
I needs to spend some time learning more oor understanding more.
Thanks for great info I was looking for this info for my mission.
It’s awesome to go to see this web site and reading the views of all mates concerning this post, while I am also eager of getting know-how.
Hi, this weekend is fastidious for me, since this occasion i am reading this
wonderful educational piece of writing here at
my home.
Hey There. I found your blog using msn. This is a really well written article.
I will be sure to bookmark it and come back to read more of your useful info.
Thanks for the post. I’ll certainly return.
For most up-to-date information you have to pay a visit web and on world-wide-web
I found this website as a finest web site for latest updates.
Excellent goods from you, man. I have understand your stuff previous
to and you’re just too excellent. I actually like what you have acquired here, really like
what you’re saying and the way in which you say it.
You make it enjoyable and you still take care of to keep
it sensible. I can not wait to read far more from you. This is
really a great web site.
Its like you read my mind! You seem to know a lot about this, like you wrote the book in it
or something. I think that you can do with some pics to drive the message home a
little bit, but instead of that, this is great blog.
An excellent read. I will definitely be back.
I delight in, cause I found exactly what I used to be looking for.
You’ve ended my 4 day lengthy hunt! God Bless you man. Have
a great day. Bye
It’s awesome to pay a visit this web site and reading the views of all friends about this post, while I am also keen of getting know-how.
Hello friends, its fantastic post regarding teachingand fully explained, keep it up all
the time.
This site was… how do you say it? Relevant!! Finally I have found something that
helped me. Thank you!
Greetings, There’s no doubt that your website may be having browser compatibility problems.
When I take a look at your blog in Safari, it looks fine but when opening in IE,
it’s got some overlapping issues. I just wanted to give you
a quick heads up! Aside from that, excellent site!
I have been exploring for a bit for any high quality articles or blog posts in this kind of house .
Exploring in Yahoo I ultimately stumbled upon this website.
Reading this info So i am happy to exhibit that I’ve a very just right uncanny feeling
I came upon just what I needed. I most no doubt will make certain to do not put out
of your mind this website and provides it a glance on a relentless basis.
Howdy! Someone in my Myspace groⲣ shaгed thus
website with us so I came to take a look.
I’m definitelｙ enjoying the inf᧐rmatiоn. I’m booҝ-marking and will
bе tweeting this to my f᧐llowers! Superb blog and great style and design. https://www.bookmark-Xray.win/judi-bola-terbaik