Question B3 One of the possible states of the electron in the Hydrogen atom is given by 1 r (r, 0, 0) = e -r/(200) sin eo, 3/2 where a is the Bohr radius. For this wavefunction, answer the following questions: a) Using the operator for the z-component of angular momentum ₁₂ = -ihǝ/ǝo, determine the value of the quantum number m₁. [4 marks] b) Using the total angular momentum operator, given below, determine the value of the quan- tum number 1. a β = -h² 1 მ sinᎾ [sin € 30 (sin 600) + 1098] sin2 c) d) [8 marks] Explain how you could have identified / and my by direct inspection of the wavefunction, and use the direct inspection method to identify the value of the quantum number n. The operator for the y-component of angular momentum is given by Ly = ih ih (-cos - მ a +cot sin o მი (i) Calculate, and show that it is not a function of o. (ii) Show that the expectation value of I, vanishes, i.e. (L) = 0. [Hint: the volume element in this case is given by dV = 2 sin 0 dr de do.] [6 marks] [7 marks]
Question B3 One of the possible states of the electron in the Hydrogen atom is given by 1 r (r, 0, 0) = e -r/(200) sin eo, 3/2 where a is the Bohr radius. For this wavefunction, answer the following questions: a) Using the operator for the z-component of angular momentum ₁₂ = -ihǝ/ǝo, determine the value of the quantum number m₁. [4 marks] b) Using the total angular momentum operator, given below, determine the value of the quan- tum number 1. a β = -h² 1 მ sinᎾ [sin € 30 (sin 600) + 1098] sin2 c) d) [8 marks] Explain how you could have identified / and my by direct inspection of the wavefunction, and use the direct inspection method to identify the value of the quantum number n. The operator for the y-component of angular momentum is given by Ly = ih ih (-cos - მ a +cot sin o მი (i) Calculate, and show that it is not a function of o. (ii) Show that the expectation value of I, vanishes, i.e. (L) = 0. [Hint: the volume element in this case is given by dV = 2 sin 0 dr de do.] [6 marks] [7 marks]
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![Question B3
One of the possible states of the electron in the Hydrogen atom is given by
1
r
(r, 0, 0) =
e
-r/(200) sin eo,
3/2
where a is the Bohr radius. For this wavefunction, answer the following questions:
a)
Using the operator for the z-component of angular momentum ₁₂ = -ihǝ/ǝo, determine
the value of the quantum number m₁.
[4 marks]
b) Using the total angular momentum operator, given below, determine the value of the quan-
tum number 1.
a
β = -h²
1 მ
sinᎾ
[sin € 30 (sin 600) + 1098]
sin2
c)
d)
[8 marks]
Explain how you could have identified / and my by direct inspection of the wavefunction,
and use the direct inspection method to identify the value of the quantum number n.
The operator for the y-component of angular momentum is given by
Ly = ih
ih (-cos -
მ
a
+cot sin o
მი
(i) Calculate, and show that it is not a function of o.
(ii) Show that the expectation value of I, vanishes, i.e. (L) = 0.
[Hint: the volume element in this case is given by dV = 2 sin 0 dr de do.]
[6 marks]
[7 marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c0d52ac-ebeb-4643-8051-a3a6486fdce4%2Feb9d53df-4e28-4ce2-bc75-60f32a5a0ff1%2F44aq6gr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question B3
One of the possible states of the electron in the Hydrogen atom is given by
1
r
(r, 0, 0) =
e
-r/(200) sin eo,
3/2
where a is the Bohr radius. For this wavefunction, answer the following questions:
a)
Using the operator for the z-component of angular momentum ₁₂ = -ihǝ/ǝo, determine
the value of the quantum number m₁.
[4 marks]
b) Using the total angular momentum operator, given below, determine the value of the quan-
tum number 1.
a
β = -h²
1 მ
sinᎾ
[sin € 30 (sin 600) + 1098]
sin2
c)
d)
[8 marks]
Explain how you could have identified / and my by direct inspection of the wavefunction,
and use the direct inspection method to identify the value of the quantum number n.
The operator for the y-component of angular momentum is given by
Ly = ih
ih (-cos -
მ
a
+cot sin o
მი
(i) Calculate, and show that it is not a function of o.
(ii) Show that the expectation value of I, vanishes, i.e. (L) = 0.
[Hint: the volume element in this case is given by dV = 2 sin 0 dr de do.]
[6 marks]
[7 marks]
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