Related questions
a) Use the known commutation rules for x, y, z, px, py and pz to show that [Ly, Lz] = ihLx.
b) Consider the spherical harmonic Y1, -1([theta], [phi]) = (1/2)*sqrt(3/2pi)*sin[theta]*e-i[phi], where [theta] and [phi] are the polar and azimuthal angles, respectively.
-> i) Express Y1, -1 in terms of cartesian coordinates.
-> ii) Show that Y1, -1 is an eigenfunction of Lz.
ci) Express the wavefunction [psi]210 for the 2pz orbital of the hydrogen atom (derived in the lectures and given in the notes) in cartesian coordinates. [Note: This involves a different spherical harmonic than in (b).]
ii) Based on this expression, show that this wavefunction satisfies the three-dimensional stationary Schrodinger equation of the hydrogen atom, and determine the corresponding energy.
I have attached the question better formatted, as well as the information from lectures referred to in part ci).
![Angular momentum in quantum mechanics is given by L = L,i+L„j+L¸k with components
(a) Use the known commutation rules for å, ŷ, 2, pz, Py, and p, to show that [L,, L-] = iħ΄.
1
3
sin(0) exp(-id), where 0 and o
2 27
(b) Consider the spherical harmonid Y1,-1(0, 6)
are the polar and azimuthal angles, respectively.
(i) Express Y1,-1 in terms of cartesian coordinates.
(ii) Show that Y1,-1 is an eigenfunction of Î..
(c) (i) Express the wavefunction p210 for the 2p, orbital of the hydrogen atom (derived in
the lectures and given in the notes) in cartesian coordinates. [Note: This involves
a different spherical harmonic than in (b).]
(ii) Based on this expression, show that this wavefunction satisfies the three-dimensional
stationary Schrödinger equation of the hydrogen atom, and determine the corre-
sponding energy.](https://content.bartleby.com/qna-images/question/7343d276-7c36-41eb-b982-4dd1d6e8ac08/003d3afb-d1f9-4832-91df-8b11c26d508a/lkxdl2s_processed.jpeg)
![The wavefunctions
Pnim(r)
Cni Yim(0, 4)L(2r/(nao))r'e¯r/(na)
T 21+1
'n-l-1
(342)
of the hydrogen atom are also called atomic orbitals. They are normalised for
Cnl = (2/(nao))+3/2 /(n – 1– 1)!/[2n(n +l)!]. The azimuthal quantum number is denoted by a symbol s for
l = 0, p for l = 1, d for l = 2, and f for l = 3. These symbols are then preceded by the principal wavenumber n, so
that orbitals are denoted by 1s, 2s, 2p, 3s, 3p, 3d etc.
\l+3/2](https://content.bartleby.com/qna-images/question/7343d276-7c36-41eb-b982-4dd1d6e8ac08/003d3afb-d1f9-4832-91df-8b11c26d508a/fhjd1sq_processed.jpeg)
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