Angular momentum in quantum mechanics is given by L = L_xi+L_yj+L_zk with components L_x= yp_z- zp_y, L_y= zp_x - xp_z, L_z = xp_y - yp_x.

a) Use the known commutation rules for x, y, z, p_x, p_yand p_z to show that [L_y, L_z] = ihL_x.

b) Consider the spherical harmonic Y_{1, -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 Y_{1, -1} in terms of cartesian coordinates.

-> ii) Show that Y_{1, -1} is an eigenfunction of L_z.

ci) Express the wavefunction [psi]₂₁₀ for the 2p_z 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).
Transcribed Image Text: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-] = iħÎ„. 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.
$The wavefunctions Pnim(r) Cni Yim(0, 4)L(2r/(nao))r'e¯r/(na) T 21+1 'n-l-1 (342) of the hydrogen atom are also called atomic orbitals. They are normalised for Cnl = (2/(nao))+3/2 /(n – 1– 1)!/[2n(n +l)!]. The azimuthal quantum number is denoted by a symbol s for l = 0, p for l = 1, d for l = 2, and f for l = 3. These symbols are then preceded by the principal wavenumber n, so that orbitals are denoted by 1s, 2s, 2p, 3s, 3p, 3d etc. \l+3/2$
Transcribed Image Text: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

Question

Angular momentum in quantum mechanics is given by L = L_xi+L_yj+L_zk with components L_x= yp_z- zp_y, L_y= zp_x - xp_z, L_z = xp_y - yp_x.

a) Use the known commutation rules for x, y, z, p_x, p_yand p_z to show that [L_y, L_z] = ihL_x.

b) Consider the spherical harmonic Y_{1, -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 Y_{1, -1} in terms of cartesian coordinates.

-> ii) Show that Y_{1, -1} is an eigenfunction of L_z.

ci) Express the wavefunction [psi]₂₁₀ for the 2p_z 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-] = iħÎ„.
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.

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

Accepted Answer

Given condition,L=Lxi^+Lyj^+Lzk^Lx=ypz-zpy , Ly=zpx-xpz , Lz=xpy-ypxwe have to…