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Angular momentum in quantum mechanics is given by L = Lxi+Lyj+Lzk with components Lx = ypz- zpy, Ly = zpx - xpz, Lz = xpy - ypx. 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).]

Angular momentum in quantum mechanics is given by L = Lxi+Lyj+Lzk with components Lx = ypz- zpy, Ly = zpx - xpz, Lz = xpy - ypx. 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).]

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Angular momentum in quantum mechanics is given by L = Lxi+Lyj+Lzk with components Lx = ypz- zpy, Ly = zpx - xpz, Lz = xpy - ypx.

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.
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-] = 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.
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
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