LINEAR COMBINATIONS OF SPHERICAL HARMONICSby Kerry Browne and Corinne ManogueCopyright 2010 Corinne Manogue
In this worksheet, you will examine combinations of the spherical
harmonics, 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First we define the 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 just as we did in the previous worksheet. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2KkYrLUYjNistRiw2JlEqQUxlZ2VuZHJlRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSM6PUYnRjYvRj1RJ25vcm1hbEYnLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVS1GIzYrLUkobWZlbmNlZEdGJDYlLUYjNiotRiw2JlEjZWxGJ0Y2RjlGPC1GQDYuUSIsRidGNkZDRkUvRkhGO0ZJRktGTUZPRlEvRlRRJjAuMGVtRicvRldRLDAuMzMzMzMzM2VtRictRiw2JlEibUYnRjZGOUY8LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGNi8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0Y4RkMvRj1RJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRmBwLUZANi5RKCYjODU5NDtGJ0Y2RkNGRUZHRklGS0ZNRk9GUUZgby9GV0Zhby1GIzYqLUklbXN1cEdGJDYlLUZlbjYlLUYjNistSSNtbkdGJDYlUSIxRidGNkZDLUZANi5RKCZtaW51cztGJ0Y2RkNGRUZHRklGS0ZNRk9GUS9GVFEsMC4yMjIyMjIyZW1GJy9GV0ZocS1GIzYoLUZqcDYlLUYsNiZRInpGJ0Y2RjlGPC1GYXE2JVEiMkYnRjZGQy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGZ29GNkZqby8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGQ0YrRmdvRjZGam9GZ3JGQ0ZfcEZhcC1JJm1mcmFjR0YkNigtRiM2KC1GZW42KUZkb0ZfcC9JK21zZW1hbnRpY3NHRiRRJGFic0YnRmFwLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGZnNGYXNGZ29GNkZqb0ZdcEZDLUYjNihGYXJGZ29GNkZqb0ZdcEZDLyUubGluZXRoaWNrbmVzc0dGY3EvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGX3QvJSliZXZlbGxlZEdGOEZkci1GQDYuUTEmSW52aXNpYmxlVGltZXM7RidGNkZDRkVGR0ZJRktGTUZPRlFGYG9GZnAtRmVuNiUtRiM2Ky1GW3M2KC1GIzYoLUZqcDYlLUZANi5RKyZQYXJ0aWFsRDtGJ0Y2RkNGRUZHRklGS0ZNRk9GUUZgb0ZmcEZfc0ZkckZnb0Y2RmpvRl1wRkMtRiM2KkYrLUYjNilGYXUtRmpwNiVGXnJGX3NGZHJGZ29GNkZqb0ZdcEZDRitGZ29GNkZqb0ZdcEZDRlt0Rl10RmB0RmJ0RmR0LUYjNiotRiw2JlEiUEYnRjZGOUY8LUZANj9RMCZBcHBseUZ1bmN0aW9uO0YnRmdvLyUlc2l6ZUdRIzE2RidGNi9GOkY4LyUqdW5kZXJsaW5lR0Y4LyUqc3Vic2NyaXB0R0Y4LyUsc3VwZXJzY3JpcHRHRjhGam8vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y4Rl1wLyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGOC9GaHJRKk5vcm1hbDI1NkYnRkNGRUZHRklGS0ZNRk9GUUZgb0ZmcC1GZW42JS1GIzYqRmluRlxvRl5yRmdvRjZGam9GXXBGQ0ZfcEZhcEZnb0Y2RmpvRl1wRkNGK0Znb0Y2RmpvRl1wRkNGX3BGYXBGZ29GNkZqb0ZnckZDRitGZ29GNkZqb0ZdcEZDRitGZ29GNkZqb0ZdcEZDRitGZ29GNkZqb0ZdcEZDLUZANi5RIjpGJ0Y2RkNGRUZHRklGS0ZNRk9GUUZTRlZGZ29GNkZqb0ZdcEZDLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2KkYrLUYjNistRiw2JlEkbnJtRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSM6PUYnRjYvRj1RJ25vcm1hbEYnLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVS1GIzYrLUkobWZlbmNlZEdGJDYlLUYjNiotRiw2JlEjZWxGJ0Y2RjlGPC1GQDYuUSIsRidGNkZDRkUvRkhGO0ZJRktGTUZPRlEvRlRRJjAuMGVtRicvRldRLDAuMzMzMzMzM2VtRictRiw2JlEibUYnRjZGOUY8LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGNi8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0Y4RkMvRj1RJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRmBwLUZANi5RKCYjODU5NDtGJ0Y2RkNGRUZHRklGS0ZNRk9GUUZgby9GV0Zhby1GIzYqLUklbXN1cEdGJDYlLUZlbjYlLUYjNiktRkA2LlEqJnVtaW51czA7RidGNkZDRkVGR0ZJRktGTUZPRlEvRlRRLDAuMjIyMjIyMmVtRicvRldGZHEtSSNtbkdGJDYlUSIxRidGNkZDRmdvRjZGam9GXXBGQ0ZfcEZhcC1JJm1mcmFjR0YkNigtRiM2KkYrLUYjNipGZG8tRkA2LlEiK0YnRjZGQ0ZFRkdGSUZLRk1GT0ZRRmNxRmVxLUZlbjYpRmRvRl9wL0krbXNlbWFudGljc0dGJFEkYWJzRidGYXAvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0Zbc0ZmckZnb0Y2RmpvLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0ZDRitGZ29GNkZqb0ZdcEZDLUYjNigtRmdxNiVRIjJGJ0Y2RkNGZ29GNkZqb0ZdcEZDLyUubGluZXRoaWNrbmVzc0dGaXEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGanMvJSliZXZlbGxlZEdGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRkA2LlExJkludmlzaWJsZVRpbWVzO0YnRjZGQ0ZFRkdGSUZLRk1GT0ZRRmBvRmZwLUkmbXNxcnRHRiQ2Iy1GW3I2KC1GIzYrLUZlbjYlLUYjNitGKy1GIzYqRmNzRmJ0RmluRmdvRjZGam9GXnNGQ0ZhckZmcUZnb0Y2RmpvRl5zRkNGX3BGYXBGYnQtRiM2KS1GZW42JS1GIzYqRmluLUZANi5RKCZtaW51cztGJ0Y2RkNGRUZHRklGS0ZNRk9GUUZjcUZlcUZkckZnb0Y2RmpvRl5zRkNGX3BGYXAtRkA2LlEiIUYnRjZGQ0ZFRkdGSUZLRk1GT0ZRL0ZUUSwwLjExMTExMTFlbUYnL0ZXRl92RmdvRjZGam9GXXBGQ0YrRmdvRjZGam9GXXBGQy1GIzYqRistRiM2LS1GZ3E2JVEiNEYnRjZGQ0ZidC1GLDYmUScmIzk2MDtGJ0Y2L0Y6RjhGQ0ZidC1GIzYpLUZlbjYlLUYjNipGaW5GYXJGZHJGZ29GNkZqb0Zec0ZDRl9wRmFwRlt2RmdvRjZGam9GXXBGQ0YrRmdvRjZGam9GXnNGQ0YrRmdvRjZGam9GXXBGQ0Zmc0Zoc0ZbdEZddEZnb0Y2RmpvRl5zRkNGK0Znb0Y2RmpvRl1wRkNGK0Znb0Y2RmpvRl1wRkNGK0Znb0Y2RmpvRl1wRkMtRkA2LlEiOkYnRjZGQ0ZFRkdGSUZLRk1GT0ZRRlNGVkZnb0Y2RmpvRl1wRkM=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2KkYrLUYjNistRiw2JlEiWUYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEjOj1GJ0Y2L0Y9USdub3JtYWxGJy8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlUtRiM2Ky1JKG1mZW5jZWRHRiQ2JS1GIzYqLUYsNiZRI2VsRidGNkY5RjwtRkA2LlEiLEYnRjZGQ0ZFL0ZIRjtGSUZLRk1GT0ZRL0ZUUSYwLjBlbUYnL0ZXUSwwLjMzMzMzMzNlbUYnLUYsNiZRIm1GJ0Y2RjlGPC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnRjYvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGOEZDL0Y9USVib2xkRicvJStmb250d2VpZ2h0R0ZgcC1GQDYuUSgmIzg1OTQ7RidGNkZDRkVGR0ZJRktGTUZPRlFGYG8vRldGYW8tRiM2LUYrLUYjNiotRiw2JlEkbnJtRidGNkY5RjwtRkA2P1EwJkFwcGx5RnVuY3Rpb247RidGZ28vJSVzaXplR1EjMTZGJ0Y2L0Y6RjgvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOEZqby8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRjhGXXAvJSlyZWFkb25seUdGOC8lKWNvbXBvc2VkR0Y4LyUqY29udmVydGVkR0Y4LyUraW1zZWxlY3RlZEdGOC8lLHBsYWNlaG9sZGVyR0Y4LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y4LyUwZm9udF9zdHlsZV9uYW1lR1EqTm9ybWFsMjU2RidGQ0ZFRkdGSUZLRk1GT0ZRRmBvRmZwRlpGZ29GNkZqb0ZdcEZDLUZANi5RMSZJbnZpc2libGVUaW1lcztGJ0Y2RkNGRUZHRklGS0ZNRk9GUUZgb0ZmcC1GIzYqLUYsNiZRJXN1YnNGJ0Y2RjlGPEZecS1GZW42JS1GIzYsRistRiM2Ky1GLDYmUSJ6RidGNkY5RjwtRkA2LlEiPUYnRjZGQ0ZFRkdGSUZLRk1GT0ZRRlNGVi1GIzYqLUYsNiZRJGNvc0YnRjZGZHFGQ0ZecS1GZW42JS1GIzYoLUYsNiZRJyYjOTUyO0YnRjZGZHFGQ0Znb0Y2RmpvRl1wRkNGX3BGYXBGZ29GNkZqb0ZdcEZDRitGZ29GNkZqb0ZdcEZDRlxvLUYjNiotRiw2JlEqQUxlZ2VuZHJlRidGNkY5RjxGXnFGWkZnb0Y2RmpvRl1wRkNGK0Znb0Y2RmpvRl1wRkNGX3BGYXBGZ29GNkZqb0ZdcEZDRl9zLUklbXN1cEdGJDYlLUZANi5RLyZFeHBvbmVudGlhbEU7RidGNkZDRkVGR0ZJRktGTUZPRlFGYG8vRldRLDAuMTExMTExMWVtRictRiM2LC1GLDYmUSJJRidGNkY5RjxGX3NGZG9GX3MtRiw2JlEnJiM5NjY7RidGNkZkcUZDRmdvRjZGam8vRl1zUSkyRH5JbnB1dEYnRkMvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmdvRjZGam9GZHZGQ0YrRmdvRjZGam9GXXBGQ0YrRmdvRjZGam9GXXBGQ0YrRmdvRjZGam9GXXBGQ0YrRmdvRjZGam9GXXBGQw==
Choose a sum of 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'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
Next, "square" the sum of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY4LUkjbWlHRiQ2I1EhRictRiM2OC1JJW1zdWJHRiQ2Ji1GLDY2USJZRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxNkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRj8vJSpzdWJzY3JpcHRHRj8vJSxzdXBlcnNjcmlwdEdGPy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y/LyUrZXhlY3V0YWJsZUdGPy8lKXJlYWRvbmx5R0Y/LyUpY29tcG9zZWRHRj8vJSpjb252ZXJ0ZWRHRj8vJStpbXNlbGVjdGVkR0Y/LyUscGxhY2Vob2xkZXJHRj8vJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRj8vJTBmb250X3N0eWxlX25hbWVHUSpOb3JtYWwyNTZGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2OC1GLDY2USJsRidGN0Y6Rj1GQEZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GXG8tSSNtb0dGJDY/USIsRidGN0Y6Rj0vRkFGP0ZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW4vRl1vUSdub3JtYWxGJy8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNjZRIm1GJ0Y3RjpGPUZARkNGRUZHRklGTEZPRlFGU0ZVRldGWUZlbkZnbkZpbkZcb0Y3RjpGPUZob0ZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GaW8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Zlbi1GZW82P1EwJkFwcGx5RnVuY3Rpb247RidGN0Y6Rj1GaG9GQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluRmlvRltwL0ZecEY/Rl9wRmFwRmNwRmVwRmdwRmlwL0ZdcUZbcS1JKG1mZW5jZWRHRiQ2Ni1GIzY4LUYsNjZRKCZ0aGV0YTtGJ0Y3RjpGPUZob0ZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GaW9GZG8tRiw2NlEmJnBoaTtGJ0Y3RjpGPUZob0ZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GaW9GN0Y6Rj1GaG9GQ0ZFRkdGSUZMRk9GUUZTRlVGV0ZZRmVuRmduRmluRmlvRjdGOkY9RmhvRkNGRUZHRklGTEZPRlFGU0ZVRldGWUZlbkZnbkZpbkZpb0Y3RjpGPUZob0ZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GaW9GK0Y3RjpGPUZob0ZDRkVGR0ZJRkxGT0ZRRlNGVUZXRllGZW5GZ25GaW5GaW8='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 on a sphere, coloring the surface with the
value of the probability density.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Try some of the following linear combinations. Note: \316\264 is an arbitrary phase constant you can set ot any value
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