Particles in a three-dimensional spherically symmetric potential by pauldirac

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physics · @pauldirac · Jun 28 '18

$0.18

Particles in a three-dimensional spherically symmetric potential

<center>![](https://cdn.steemitimages.com/DQmcGUq2kvZwZsBtLDfokSrKXhvhhCBGVfDEU21MrW82h8N/image.png)</center>
**Disclaimer: I lay no claim to the originality of the basic ideas expounded here. Its distinctness lies more in its scope and in the detail of its exposition.**

# Particles in a 3D Symmetric Potential
In a three dimensional spherically symmetric potential ![](https://cdn.steemitimages.com/DQmVrcXqw7K6WGxYRoCbGwQNSV6pR2tDTDvtCsSmztNcHqy/image.png) the time-independent Schrodinger equation for stationary states of energy E is 
<center>![](https://cdn.steemitimages.com/DQmbV2w4NqspdtfCVTkSQQqHM5Z9BhmwJFHW9Nw8kZhFwiQ/image.png)</center>

This Schrodinger equation in polar coordinates is expressed by
<center>![](https://cdn.steemitimages.com/DQmbhtg3mtSioFNSP1Bxr64XbbbkNsXuqtpokgeXg14FSAD/image.png)</center>

We multiply this equation by ![](https://cdn.steemitimages.com/DQmaJm16adTtShFJsa8tJEoXKggy6tZsu9YCQFdDmUn2GaA/image.png)
<center>![](https://cdn.steemitimages.com/DQmY5pvvrhJZykHAqjeu3JU4spzs1aie3SX3CqUQTyoTqzE/image.png)</center>

We then, substitute this wavefunction, 
<center>![](https://cdn.steemitimages.com/DQmYnJEiHjH4ByHmBBbfDxqX7sKWzAvQ8FwdV43HEcsLcnX/image.png)</center>

We have
<center>![](https://cdn.steemitimages.com/DQmUE7wVD18cCDTBuUJxNmTPkMKCNUiDbeDuauQBg5UzFqH/image.png)</center>

Note that the left side is only a function of r, meaning we can shift the partial derivative to a total derivative, 
<center>![](https://cdn.steemitimages.com/DQmZPEvkinLtQhCoHbtDVonzWKVR79LwgD35toxVJakL1nj/image.png)</center>

On the other hand, the left side is a function of $\theta$ and $\varphi$ only, 
<center>![](https://cdn.steemitimages.com/DQmeskZRJmdcxGzwomiWEqBZ1owjieGbBqjLafddQB4JWGc/image.png)</center>

Note also that they must be equal to a constant because of the different variable dependence. They must be equal to the same constant.

Remember this equation: 
<center>![](https://cdn.steemitimages.com/DQmVEL16JKfpbzyR371Na6NFKa4Yp4WfrYvbaxHX6Canbxv/image.png)</center>

Thus, the angular part can be written as,
<center>![](https://cdn.steemitimages.com/DQmPcX7rweb8wXDwGHU5eL51NcgKLaPo74MRaeUZ3m1qwfv/image.png)</center>

From the previous section, we've encountered this operator and found that its eigenvalue spectrum are
<center>![](https://cdn.steemitimages.com/DQmaLTA49R5e6VxJwikUommoGiLAMYtw8SJCgzzV4RLboK4/image.png)</center>

Now, let's consider the radial part of the equation, 
<center>![](https://cdn.steemitimages.com/DQmQrdVtjCEUhgGifMcia9MoNkNekZtWRuMexeHPAofQYoZ/image.png)</center>

We can substitute our value of ![](https://cdn.steemitimages.com/DQmQhHqhTdzCaLtoZC2sGx7BZyH7SteyHvGMS95RvkoQe1M/image.png) into this equation, 
<center>![](https://cdn.steemitimages.com/DQmWenVRTTNDMm7NF2iMru9PXnS4PovpBHFMF6Rd7RCM63p/image.png)</center>

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`permlink`	steemitboard-notify-pauldirac-20181106t142213000z
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@steemitboard · Nov 25 '18

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<table><tr><td><a href="https://steemit.com/steemfest/@steemitboard/meet-the-steemians-contest-the-results-the-winners-and-the-prizes"><img src="https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmeLukvNFRsa7RURqsFpiLGEZZD49MiU52JtWmjS5S2wtW/image.png"></a></td><td><a href="https://steemit.com/steemfest/@steemitboard/meet-the-steemians-contest-the-results-the-winners-and-the-prizes">Meet the Steemians Contest - The results, the winners and the prizes</a></td></tr></table>

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@steemitboard · Dec 13 '18

Congratulations @pauldirac! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

<table><tr><td>https://steemitimages.com/60x70/http://steemitboard.com/@pauldirac/votes.png?201812130640</td><td>You made more than 1750 upvotes. Your next target is to reach 2000 upvotes.</td></tr>
</table>

<sub>_[Click here to view your Board of Honor](https://steemitboard.com/@pauldirac)_</sub>
<sub>_If you no longer want to receive notifications, reply to this comment with the word_ `STOP`</sub>



> Support [SteemitBoard's project](https://steemit.com/@steemitboard)! **[Vote for its witness](https://v2.steemconnect.com/sign/account-witness-vote?witness=steemitboard&approve=1)** and **get one more award**!

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`created`	2018-12-13 07:33:57
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@steemitboard · Jan 1 '19

Congratulations @pauldirac! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

<table><tr><td>https://steemitimages.com/60x70/http://steemitboard.com/@pauldirac/votes.png?201901010432</td><td>You made more than 2000 upvotes. Your next target is to reach 3000 upvotes.</td></tr>
</table>

<sub>_[Click here to view your Board](https://steemitboard.com/@pauldirac)_</sub>
<sub>_If you no longer want to receive notifications, reply to this comment with the word_ `STOP`</sub>



**Do not miss the last post from @steemitboard:**
<table><tr><td><a href="https://steemit.com/christmas/@steemitboard/christmas-challenge-send-a-gift-to-to-your-friends-the-party-continues"><img src="https://steemitimages.com/64x128/http://i.cubeupload.com/kf4SJb.png"></a></td><td><a href="https://steemit.com/christmas/@steemitboard/christmas-challenge-send-a-gift-to-to-your-friends-the-party-continues">Christmas Challenge - The party continues</a></td></tr></table>

> Support [SteemitBoard's project](https://steemit.com/@steemitboard)! **[Vote for its witness](https://v2.steemconnect.com/sign/account-witness-vote?witness=steemitboard&approve=1)** and **get one more award**!

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`author`	steemitboard
`permlink`	steemitboard-notify-pauldirac-20190101t052235000z
`category`	physics
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`created`	2019-01-01 05:22:33
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`percent_hbd`	10,000
`post_id`	77,694,326
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@steemitboard · May 17 '19

Congratulations @pauldirac! You received a personal award!

<table><tr><td>https://steemitimages.com/70x70/http://steemitboard.com/@pauldirac/birthday1.png</td><td>Happy Birthday! - You are on the Steem blockchain for 1 year!</td></tr></table>

<sub>_You can view [your badges on your Steem Board](https://steemitboard.com/@pauldirac) and compare to others on the [Steem Ranking](http://steemitboard.com/ranking/index.php?name=pauldirac)_</sub>


**Do not miss the last post from @steemitboard:**
<table><tr><td><a href="https://steemit.com/japanese/@steemitboard/new-japanese-speaking-community-steem-meetup-badge"><img src="https://steemitimages.com/64x128/https://cdn.steemitimages.com/DQmRWbAjbeETEaqSPLcpwYX1JN5pZhdPffv4q6DaBs6xvZm/image.png"></a></td><td><a href="https://steemit.com/japanese/@steemitboard/new-japanese-speaking-community-steem-meetup-badge">New  japanese speaking community Steem Meetup badge</a></td></tr></table>

###### [Vote for @Steemitboard as a witness](https://v2.steemconnect.com/sign/account-witness-vote?witness=steemitboard&approve=1) to get one more award and increased upvotes!

properties (22)