Quantum Mechanics: Schr\303\266dinger vs Heisenberg picturePascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 (1) Laboratoire PhLAM, UMR CNRS 8523, Universit\303\251 Lille 1, F-59655, France(2) MaplesoftWithin the Schr\303\266dinger picture of Quantum Mechanics, the time evolution of the state of a system, represented by a Ket LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2L1EifGdyRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGSS1GLDYvUSJ+RidGL0YyRjVGOEY6L0Y9RjFGP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS0ZRLUkjbWlHRiQ2J1EnJiM5Njg7RicvJSdpdGFsaWNHRjFGL0YyRjUtSShtZmVuY2VkR0YkNiYtRiM2JS1GVDYnUSJ0RicvRlhGPkYvRjIvRjZRJ2l0YWxpY0YnRi9GNUYvRjJGNUZMLUYsNi9RNCZSaWdodEFuZ2xlQnJhY2tldDtGJ0YvRjJGNS9GOUY+RjpGPEY/RkFGQ0ZFRlBGUi1GVDYjUSFGJ0YvRjU=, is determined by Schr\303\266dinger's equation:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUYjNiwtSSNtbkdGJDYkUSJpRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1JI21pR0YkNiVRJyZoYmFyO0YnLyUnaXRhbGljR0Y6RjFGNC1JKm1jb21wbGV0ZUdGJDYkLUYjNiotSSZtZnJhY0dGJDYoLUY1Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjFGOEY7Rj1GP0ZBRkNGRUZHRkotRiM2KUZaLUZNNiVRInRGJy9GUVEldHJ1ZUYnL0YyUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRjovJSlyZWFkb25seUdGXW8vJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0YxLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZfcC8lKWJldmVsbGVkR0Y6LUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmlwLyUqbGluZWJyZWFrR1ElYXV0b0YnLUZTNiQtSShtZmVuY2VkR0YkNiYtRiM2Ki1GZXA2JkZncC9GW3FRJjAuM2VtRidGXXFGX3EtSSVtc3ViR0YkNiUtRk02JVEmJnBzaTtGJ0ZQRjEtRmVxNiZGaW5GMS8lJW9wZW5HUSFGJy8lJmNsb3NlR0Znci8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRmlxRmBvRmNvRmVvRmdvRjFGMS9GZnJRKSZ2ZXJiYXI7RicvRmlyUScmcmFuZztGJy1GIzYqLUZNNiVRJEtldEYnRlxvRl5vLUY1Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjFGOEY7Rj1GP0ZBRkNGRUZHRkotRmVxNiQtRiM2KEZgci1GNTYtUSIsRidGMUY4L0Y8Rl1vRj1GP0ZBRkNGRUZHL0ZLUSwwLjMzMzMzMzNlbUYnRmluRmNvRmdvRjFGMUZgb0Zjb0Zlb0Znb0YxRmBvRmNvRmVvRmdvRjEtRiM2Ki1GWDYoLUY1Ni1RKyZQYXJ0aWFsRDtGJ0YxRjhGO0Y9Rj9GQUZDRkVGR0ZKLUYjNilGZ3RGaW5GYG9GY29GZW9GZ29GMUZqb0ZdcEZgcEZicEZkcEZicUZgb0Zjb0Zlb0Znb0YxRmBvRmNvRmVvRmdvRjEtRjU2LVEpJmVxdWFscztGJ0YxRjhGO0Y9Rj9GQUZDRkUvRkhRLDAuMjc3Nzc3OGVtRicvRktGYHUtRiM2Ki1GTTYmUSJIRidGXG8vRmFvUSxbMTI4LDEyOCwwXUYnRl5vRjRGYnFGYG9GY29GZW9GZ29GMUZgb0Zjb0Zlb0Znb0Yxwhere H, the Hamiltonian, as well as the quantum operators LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRi82JVEiU0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGNQ== representing observable quantities, are all time-independent.Within the Heisenberg picture, a Ket 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 representing the state of the system does not evolve with time, but the operators 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 observable quantities, and through them the Hamiltonian H, do.Problem: Departing from Schr\303\266dinger's equation, a) Show that the expected value of a physical observable in Schr\303\266dinger's and Heisenberg's representations is the same, i.e. that 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b) Show that the evolution equation of an observable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYmLUkjbWlHRiQ2KFEiT0YnLyUlc2l6ZUdRIzEyRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRi82KFEiSEYnRjIvRjZGOkY4RjsvRj9RJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GLzYjUSFGJ0YyRjgvRjxRKTJEfklucHV0RidGPg== in Heisenberg's picture, equivalent to Schr\303\266dinger's equation, is given by: 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where in the right-hand-side we see the commutator of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYvNiZRIkhGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GLzYjUSFGJ0Y1LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y3 with the Hamiltonian of the system.
Solution: Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYvNiZRIlNGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y3 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYvNiZRIkhGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y3 respectively be operators representing one and the same observable quantity in Schr\303\266dinger's and Heisenberg's pictures, and H be the operator representing the Hamiltonian of a physical system. All of these operators are Hermitian. So we start by setting up the framework for this problem accordingly, including that the time t and Planck's constant are real. To automatically combine powers of the same base (happening frequently in what follows) we also set combinepowersofsamebase = true. The following input/output was obtained using the latest Physics update (Aug/31/2016) distributed on the Maplesoft R&D Physics webpage.QyYtSSV3aXRoRzYiNiNJKFBoeXNpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiItSSppbnRlcmZhY2VHRig2Iy9JLmltYWdpbmFyeXVuaXRHRiVJImlHRiVGKw==LUkmU2V0dXBHNiUlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSShQaHlzaWNzRzYkRiVJKF9zeXNsaWJHRigvSTFoaWRlbW9kdWxlcHJlZml4R0YoSSV0cnVlR0YlNiYvSTNoZXJtaXRpYW5vcGVyYXRvcnNHRig8JUkiSEdGKEklT19fSEdGKEklT19fU0dGKC9JLHJlYWxvYmplY3RzR0YoPCRJJyZoYmFyO0dGKEkidEdGKC9JOGNvbWJpbmVwb3dlcnNvZnNhbWViYXNlR0YoRi4vSTVtYXRoZW1hdGljYWxub3RhdGlvbkdGKEYuLet's consider Schr\303\266dinger's 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Now, H is time-independent, so can be formally solved: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0YxLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JS1GLDYmUSJ0RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjQtRiw2I1EhRidGMkY0 is obtained from the solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1JI21uR0YkNiVRIjBGJ0YyRjRGMkY0RjJGNEYyRjQ= at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8, as follows: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 check that is a solution of , substitute it in :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Next, to relate the Schr\303\266dinger and Heisenberg representations of an Hermitian operator O representing an observable physical quantity, recall that the value expected for this quantity at time t during a measurement is given by the mean value of the corresponding operator (i.e., bracketing it with the state of the system LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSptY29tcGxldGVHRiQ2JC1JKG1mZW5jZWRHRiQ2Jy1GIzYqLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjNlbUYnLyUmZGVwdGhHRjwvJSpsaW5lYnJlYWtHUSVhdXRvRictSSVtc3ViR0YkNiUtRiw2JlEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZNLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GMzYnLUYsNiZRInRGJy9GTFEldHJ1ZUYnRk4vRlFRJ2l0YWxpY0YnRk5GUC8lJW9wZW5HRi4vJSZjbG9zZUdGLi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjcvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRidGTi8lKXJlYWRvbmx5R0ZZLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGUEZORlAvRmduUSkmdmVyYmFyO0YnL0ZpblEnJnJhbmc7RictRiM2Ki1GLDYmUSRLZXRGJ0ZYRk5GWi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRlAvJSZmZW5jZUdGTS8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZicS1GMzYlLUYjNihGSC1GX3A2LlEiLEYnRk5GUEZicC9GZXBGWUZmcEZocEZqcEZccUZecUZgcS9GZHFRLDAuMzMzMzMzM2VtRidGVUZORmJvRlBGTkZQRl1vRk5GYG9GYm9GUEZdb0ZORmBvRmJvRlA=). So let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYvNiZRIlNGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1Rjc= be an observable in the Schr\303\266dinger picture: its mean value is obtained by bracketing the operator with equation :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The composed operator within the bracket on the right-hand-side is the operator O in Heisenberg's picture, 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Analogously, inverting this equation,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 an aside to the problem, we note from these two equations, and since the operator 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 is unitary (because H is Hermitian), that the switch between Schr\303\266dinger's and Heisenberg's pictures is accomplished through a unitary transformation.Inserting now this value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYvNiZRIlNGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GLzYjUSFGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJ0Y1LyUpcmVhZG9ubHlHRj4vJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Y3 from in the right-hand-side of , we get the answer to item a)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where, on the left-hand-side, the Ket representing the state of the system is evolving with time (Schr\303\266dinger's picture), while on the the right-hand-side the Ket 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is constant and it is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYvNiZRIkhGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYmLUYvNiZRInRGJ0Y9RjVGP0Y1LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y3RjVGNy1GLzYjUSFGJ0Y1Rk9GNw==, the operator representing an observable physical quantity, that evolves with time (Heisenberg picture). As expected, both pictures result in the same expected value for the physical quantity represented by LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUSxbMTI4LDEyOCwwXUYnLyUrZXhlY3V0YWJsZUdGLi8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=. To complete item b), the derivation of the evolution equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYvNiZRIkhGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYmLUYvNiZRInRGJ0Y9RjVGP0Y1LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y3RjVGNy1GLzYjUSFGJ0Y1Rk9GNw==, we take the time derivative of the equation :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 rewrite this equation in terms of the commutator 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, it suffices to re-order the product H 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 placing the exponential first: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Finally, to express the right-hand-side in terms of LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYpLUklbXJvd0dGJDYoLUYvNiotRi82Ki1GIzYmLUkjbWlHRiQ2JlEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRj0vJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUY4NiZRIkhGJy9GPFEldHJ1ZUYnRj4vRkFRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRkAvJSZmZW5jZUdGPS8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zeby1GLDYlLUY4NiZRInRGJ0ZGRj5GSEY+RkAvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRidGPi8lKXJlYWRvbmx5R0ZHLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGQC1GUTYuUSgmY29tbWE7RidGPkZARlQvRldGR0ZYRlpGZm5GaG5Gam5GXG8vRmBvUSwwLjMzMzMzMzNlbUYnLUY4NidGRUZGL0Znb1EsWzEyOCwxMjgsMF1GJ0Y+RkhGZm9GPkZpb0ZbcEZARmZvRj5GaW9GW3BGQEY+RkAvRk5RMlBoeXNpY3NDb21tdXRhdG9yRicvJSVvcGVuR1EnJmxzcWI7RicvJSZjbG9zZUdRJyZyc3FiO0YnRmhwLUZRNi9RKCZtaW51cztGJ0Y+RkBGaHBGVEZWRlhGWkZmbkZobkZqbi9GXW9RLDAuMjIyMjIyMmVtRicvRmBvRmRxRkpGaHA= instead of 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, we take the commutator of the equation with the 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Combining these two expressions, we arrive at the expected result for b), the evolution equation of a given observable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2J1EiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictRi82J1EiSEYnL0YzUSV0cnVlRidGNS9GOFEnaXRhbGljRidGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjotRi82I1EhRidGNS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGNw== in Heisenberg's 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