{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T14:04:17+00:00","modifiedTime":"2022-10-06T20:19:55+00:00","timestamp":"2022-10-06T21:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"Finding x, y, and z Equations for Three-Dimensional Free Particle Problems","strippedTitle":"finding x, y, and z equations for three-dimensional free particle problems","slug":"finding-the-x-y-and-z-equations-for-three-dimensional-free-particle-problems","canonicalUrl":"","seo":{"metaDescription":"In quantum physics, learn how to find the x, y, and z equations for three-dimensional free particle problems.","noIndex":0,"noFollow":0},"content":"At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397870.image0.png\" alt=\"image0.png\" width=\"140\" height=\"43\" />\r\n\r\nYou can write its general solution as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397871.image1.png\" alt=\"image1.png\" width=\"456\" height=\"156\" />\r\n\r\nwhere Ay and Az are constants.\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397872.image2.png\" alt=\"image2.png\" width=\"180\" height=\"27\" />\r\n\r\nyou get this for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397873.image3.png\" alt=\"image3.png\" width=\"229\" height=\"100\" />\r\n\r\nwhere A= Ax Ay Az.\r\n\r\nThe part in the parentheses in the exponent is the dot product of the vectors\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397874.image4.png\" alt=\"image4.png\" width=\"97\" height=\"19\" />\r\n\r\nThat is, if the vector a = (ax, ay, az) in terms of components and the vector b = (bx, by, bz), then the dot product of a and b is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397875.image5.png\" alt=\"image5.png\" width=\"176\" height=\"29\" />\r\n\r\nSo here’s how you can rewrite the\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397876.image6.png\" alt=\"image6.png\" width=\"127\" height=\"63\" />","description":"At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397870.image0.png\" alt=\"image0.png\" width=\"140\" height=\"43\" />\r\n\r\nYou can write its general solution as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397871.image1.png\" alt=\"image1.png\" width=\"456\" height=\"156\" />\r\n\r\nwhere Ay and Az are constants.\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397872.image2.png\" alt=\"image2.png\" width=\"180\" height=\"27\" />\r\n\r\nyou get this for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397873.image3.png\" alt=\"image3.png\" width=\"229\" height=\"100\" />\r\n\r\nwhere A= Ax Ay Az.\r\n\r\nThe part in the parentheses in the exponent is the dot product of the vectors\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397874.image4.png\" alt=\"image4.png\" width=\"97\" height=\"19\" />\r\n\r\nThat is, if the vector a = (ax, ay, az) in terms of components and the vector b = (bx, by, bz), then the dot product of a and b is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397875.image5.png\" alt=\"image5.png\" width=\"176\" height=\"29\" />\r\n\r\nSo here’s how you can rewrite the\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397876.image6.png\" alt=\"image6.png\" width=\"127\" height=\"63\" />","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" Dr. Steven Holzner has written more than 40 books about physics and programming. He was a contributing editor at PC Magazine and was on the faculty at both MIT and Cornell. He has authored Dummies titles including Physics For Dummies and Physics Essentials For Dummies. Dr. Holzner received his PhD at Cornell. 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Finding x, y, and z Equations for Three-Dimensional Free Particle Problems

By: Steven Holzner and

Updated: 10-06-2022

From The Book: Quantum Physics For Dummies

Quantum Physics For Dummies

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At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle,

You can write its general solution as

where A_y and A_z are constants.

Because

you get this for

where A= A_x A_y A_z.

The part in the parentheses in the exponent is the dot product of the vectors

That is, if the vector a = (a_x, a_y, a_z) in terms of components and the vector b = (b_x, b_y, b_z), then the dot product of a and b is

So here’s how you can rewrite the

About This Article

This article is from the book:

Quantum Physics For Dummies ,

About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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