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1. Commutators For the electron around the nucleus, classical physics defines the angular momentum $\vec{L}$ as the vector product of the position $\vec{r}$ and linear momentum $\vec{p}$ vectors such that: $\vec{L} \begin{pmatrix} y p_z - z p_y \ z p_x - x p_z \ x p_y - y p_x \end{pmatrix}$ with $\vec{p} = -i\hbar \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix}$ a. Calculate the following commutators. Use Cartesian coordinates (x, y, z). (i) $[L_z, L_x]$ (ii) Without calculation infer $[L_x, L_y]$ and $[L_y, L_z]$ b. Is it possible to know all the components of the angular momentum? Justify.

          1. Commutators
For the electron around the nucleus, classical physics defines the angular momentum $\vec{L}$ as the vector
product of the position $\vec{r}$ and linear momentum $\vec{p}$ vectors such that:
$\vec{L} \begin{pmatrix} y p_z - z p_y \ z p_x - x p_z \ x p_y - y p_x \end{pmatrix}$ with $\vec{p} = -i\hbar \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix}$
a. Calculate the following commutators. Use Cartesian coordinates (x, y, z).
(i) $[L_z, L_x]$
(ii) Without calculation infer $[L_x, L_y]$ and $[L_y, L_z]$
b. Is it possible to know all the components of the angular momentum? Justify.

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1. Commutators
For the electron around the nucleus, classical physics defines the angular momentum L⃗ as the vector
product of the position r⃗ and linear momentum p⃗ vectors such that:
L⃗
< p m a t r i x > with p⃗ = -iħ
< p m a t r i x >
a. Calculate the following commutators. Use Cartesian coordinates (x, y, z).
(i) [Lz, Lx]
(ii) Without calculation infer [Lx, Ly] and [Ly, Lz]
b. Is it possible to know all the components of the angular momentum? Justify.

Added by Angel H.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Lainey Roebuck

11/28/2022

Step 1

Step 1: Calculate the commutator [Lz, Lx] Using the commutation relation [Lz, Lx] = LzLx - LxLz, we can expand this as follows: [Lz, Lx] = (ypz - zpy)(xpy - ypx) - (xpy - ypx)(ypz - zpy) [Lz, Lx] = ypzxpy - ypxypz - zpyxpy + zpyypx - xpyypz + ypxzpy [Lz, Lx] = Show more…

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Commutators For the electron around the nucleus, classical physics defines the angular momentum vec(L) as the vector product of the position vec(r) and linear momentum vec(p) vectors such that: hat(L)|[hat(y)hat(p)_(z)-hat(z)hat(p)_(y)],[hat(z)hat(p)_(x)-hat(x)hat(p)_(z)],[hat(x)hat(p)_(y)-hat(y)hat(p)_(x)]|{(:[-iℏ(del)/(delx)]),(-iℏ(del)/(dely)),(-iℏ(del)/(delz)):} a. Calculate the following commutators. Use Cartesian coordinates (x,y,z). (i) hat(L)_(z),hat(L)_(x) (ii)Without calculation infer hat(L)_(x),hat(L)_(y) and hat(L)_(y),hat(L)_(z) b. Is it possible to know all the components of the angular momentum? Justify. 1.Commutators For the electron around the nucleus,classical physics defines the angular momentum L as the vector product of the position r and linear momentum p vectors such that: |ypz-2py ax ay xpy-ypx az a.Calculate the following commutators.Use Cartesian coordinates (x,y,z. (i)[Cz,Lx] (ii)Without calculation infer [Lx,Ly] and [Ly,Lz] b.Is it possible to know all the components of the angular momentum? Justify

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