1. Rotating Quantum particle: Consider a free quantum particle of mass m rotating in a circular path of constant radius, R, about the z - aris; In order to obtain this particle's wavefunction and to extract physical quantities answer the following: (a) Show that in polar coordinates the Laplacian is given by + where r = Va2 + y², o = arctan(y/x) and (x, y) are the usual Cartesian coordi- nates. (b) Show that the TISE for the particle described above is thus given by h? d 2mR do? Ev, and find the solution (c) Given (6) = »(o+27), use your solution in (b) to show that En 2mR where n is an interger You may need to use: et2rin = 1, cos a - cos 3 = -2 sin () sin (*) and/or sin a – sin 3 = 2 cos () sin (5) COS (d) Clearly the energy in equation (3) is purely kinetic. Use the formula for kinetic energy in terms of linear momentum, p, together with the formula, L = pR, to show that the angular momentum, L, is quantized and is given by Bohr's formula: [3] L = nh
1. Rotating Quantum particle: Consider a free quantum particle of mass m rotating in a circular path of constant radius, R, about the z - aris; In order to obtain this particle's wavefunction and to extract physical quantities answer the following: (a) Show that in polar coordinates the Laplacian is given by + where r = Va2 + y², o = arctan(y/x) and (x, y) are the usual Cartesian coordi- nates. (b) Show that the TISE for the particle described above is thus given by h? d 2mR do? Ev, and find the solution (c) Given (6) = »(o+27), use your solution in (b) to show that En 2mR where n is an interger You may need to use: et2rin = 1, cos a - cos 3 = -2 sin () sin (*) and/or sin a – sin 3 = 2 cos () sin (5) COS (d) Clearly the energy in equation (3) is purely kinetic. Use the formula for kinetic energy in terms of linear momentum, p, together with the formula, L = pR, to show that the angular momentum, L, is quantized and is given by Bohr's formula: [3] L = nh
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Rotating Quantum particle: Consider a free quantum particle of mass m rotating
in a circular path of constant radius, R, about the z − axis; In order to obtain this
particle’s wavefunction and to extract physical quantities answer the followin:
![1. Rotating Quantum particle: Consider a free quantum particle of mass m rotating
in a circular path of constant radius, R, about the z - axis; In order to obtain this
particle's wavefunction and to extract physical quantities answer the following:
(a) Show that in polar coordinates the Laplacian is given by
1 02
r dr
where r = Væ² + y?, ¢ = arctan(y/x) and (r, y) are the usual Cartesian coordi-
nates.
(b) Show that the TISE for the particle described above is thus given by
= E,
2m R2 do?
and find the solution
(c) Given (6) = v(0+ 27), use your solution in (b) to show that
E, =
2mR?
where n is an interger
You may need to use: e+2min = 1, cos a - cos B = -2 sin (+2) sin (,) and/or
sin a – sin 3 = 2 cos () sin (5)
(d) Clearly the energy in equation (3) is purely kinetic. Use the formula for kinetic
energy in terms of linear momentum, p, together with the formula, L = pR, to
show that the angular momentum, L, is quantized and is given by Bohr's formula:
[3]
L = nh](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a3001f1-b73c-4fdc-bda8-e664a9a2eee5%2F428e6be5-bda7-4214-9e1b-b1ed0585b4cd%2F3l076q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Rotating Quantum particle: Consider a free quantum particle of mass m rotating
in a circular path of constant radius, R, about the z - axis; In order to obtain this
particle's wavefunction and to extract physical quantities answer the following:
(a) Show that in polar coordinates the Laplacian is given by
1 02
r dr
where r = Væ² + y?, ¢ = arctan(y/x) and (r, y) are the usual Cartesian coordi-
nates.
(b) Show that the TISE for the particle described above is thus given by
= E,
2m R2 do?
and find the solution
(c) Given (6) = v(0+ 27), use your solution in (b) to show that
E, =
2mR?
where n is an interger
You may need to use: e+2min = 1, cos a - cos B = -2 sin (+2) sin (,) and/or
sin a – sin 3 = 2 cos () sin (5)
(d) Clearly the energy in equation (3) is purely kinetic. Use the formula for kinetic
energy in terms of linear momentum, p, together with the formula, L = pR, to
show that the angular momentum, L, is quantized and is given by Bohr's formula:
[3]
L = nh
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