P7F.9 In this problem you will establish the commutation relations, given in eqn 7E. 14, between the operators for the x-, y-, and z-components of angular momentum, which are defined in eqn 7F.13. In order to manipulate the operators correctly it is helpful to imagine that they are acting on some arbitrary function f: it does not matter what fis, and at the end of the proof it is simply removed. Consider [,,1,] = ,-11. Consider the effect of the first term on some arbitrary function fand evaluate A D -x dx se The next step is to multiply out the parentheses, and in doing so care needs to be taken over the order of operations. (b) Repeat the procedure for the other term in the commutator, 1,1, f. (c) Combine the results from (a) and (b) so as to evaluate l f-11f;you should find that many of the terms cancel. Confirm that the final expression you have is indeed iħl_f, where l̟ is given in eqn 7F.13. (d) The definitions in eqn 7E.13 are related to one another by cyclic permutation of the x, y, and z. That is, by making the permutation x→y, yz, and z→x, you can move from one definition to the next: confirm that this is so. (e) The same cyclic permutation can be applied to the commutators of these operators. Start with [!, 1,]=ihl, and show that cyclic permutation generates the other two commutators in eqn 7F.14.
P7F.9 In this problem you will establish the commutation relations, given in eqn 7E. 14, between the operators for the x-, y-, and z-components of angular momentum, which are defined in eqn 7F.13. In order to manipulate the operators correctly it is helpful to imagine that they are acting on some arbitrary function f: it does not matter what fis, and at the end of the proof it is simply removed. Consider [,,1,] = ,-11. Consider the effect of the first term on some arbitrary function fand evaluate A D -x dx se The next step is to multiply out the parentheses, and in doing so care needs to be taken over the order of operations. (b) Repeat the procedure for the other term in the commutator, 1,1, f. (c) Combine the results from (a) and (b) so as to evaluate l f-11f;you should find that many of the terms cancel. Confirm that the final expression you have is indeed iħl_f, where l̟ is given in eqn 7F.13. (d) The definitions in eqn 7E.13 are related to one another by cyclic permutation of the x, y, and z. That is, by making the permutation x→y, yz, and z→x, you can move from one definition to the next: confirm that this is so. (e) The same cyclic permutation can be applied to the commutators of these operators. Start with [!, 1,]=ihl, and show that cyclic permutation generates the other two commutators in eqn 7F.14.
Physical Chemistry
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Chapter10: Introduction To Quantum Mechanics
Section: Chapter Questions
Problem 10.26E: For a particle in a state having the wavefunction =2asinxa in the range x=0toa, what is the...
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![P7F.9 In this problem you will establish the commutation relations, given
in eqn 7E. 14, between the operators for the x-, y-, and z-components of
angular momentum, which are defined in eqn 7F.13. In order to manipulate
the operators correctly it is helpful to imagine that they are acting on some
arbitrary function f: it does not matter what fis, and at the end of the proof it
is simply removed. Consider [,,1,] = ,-11. Consider the effect of the first
term on some arbitrary function fand evaluate
A
D
-x
dx
se
The next step is to multiply out the parentheses, and in doing so care needs
to be taken over the order of operations. (b) Repeat the procedure for the
other term in the commutator, 1,1, f. (c) Combine the results from (a) and (b)
so as to evaluate l f-11f;you should find that many of the terms cancel.
Confirm that the final expression you have is indeed iħl_f, where l̟ is given
in eqn 7F.13. (d) The definitions in eqn 7E.13 are related to one another by](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2557f741-816d-4c4c-bf5c-4ac0c6878167%2Fa77064a8-2aae-4953-86d8-64cee3a9776b%2Fgmu48cp.png&w=3840&q=75)
Transcribed Image Text:P7F.9 In this problem you will establish the commutation relations, given
in eqn 7E. 14, between the operators for the x-, y-, and z-components of
angular momentum, which are defined in eqn 7F.13. In order to manipulate
the operators correctly it is helpful to imagine that they are acting on some
arbitrary function f: it does not matter what fis, and at the end of the proof it
is simply removed. Consider [,,1,] = ,-11. Consider the effect of the first
term on some arbitrary function fand evaluate
A
D
-x
dx
se
The next step is to multiply out the parentheses, and in doing so care needs
to be taken over the order of operations. (b) Repeat the procedure for the
other term in the commutator, 1,1, f. (c) Combine the results from (a) and (b)
so as to evaluate l f-11f;you should find that many of the terms cancel.
Confirm that the final expression you have is indeed iħl_f, where l̟ is given
in eqn 7F.13. (d) The definitions in eqn 7E.13 are related to one another by
![cyclic permutation of the x, y, and z. That is, by making the permutation x→y,
yz, and z→x, you can move from one definition to the next: confirm that
this is so. (e) The same cyclic permutation can be applied to the commutators
of these operators. Start with [!, 1,]=ihl, and show that cyclic permutation
generates the other two commutators in eqn 7F.14.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2557f741-816d-4c4c-bf5c-4ac0c6878167%2Fa77064a8-2aae-4953-86d8-64cee3a9776b%2Fc64pwzy.png&w=3840&q=75)
Transcribed Image Text:cyclic permutation of the x, y, and z. That is, by making the permutation x→y,
yz, and z→x, you can move from one definition to the next: confirm that
this is so. (e) The same cyclic permutation can be applied to the commutators
of these operators. Start with [!, 1,]=ihl, and show that cyclic permutation
generates the other two commutators in eqn 7F.14.
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