Quantum Unit 1 PDF

# Linear Algebra ## Complex Vector Space - **C = a + ib** - **C* = a - ib** Complex Conjugate of each other - **<A> = Σ Cixi ** ## Linear Combination of Vectors - If **A1, A2, A3, -- An** = set of vectors in vector space **V** - **C1A1 + C2A2 +--+ CnAn = Σ C1Ai** ## Linear Dependence - **C1 ≠ C2 = C3 = ... = Cn = 0** - **C1 = C2 = C3 = ... = Cn = 0** ## Transformation of Vector Space - **Inner product** - **scalar product** - **<A.B> = |A| |B| cos θ** - If **θ=0** then **cos θ = 1** - **<A.B> = |A| |B|** - **<A.B> = AxBx + AyBy + AzBz** - **Outer product** - **cross product** - **A x B = |A||B| sin θ** ## Bases and Dimension - **A= ai + bj + ck** - **(î, ĵ, k)** - Unit Vector ## Completeness of Vector Space: - **(Clauser property of vector space)** - **(Orthogonal vector)** - **Ui, Uj** - **Ui. Uj = 0 (i ≠ j)** - **Ui. Uj = 1 (i = j)** - Then **vector is Normalise** - **S = (Φn)** - **<Ψ| Φn> = C1<Φ1| Φn> + C2<Φ2| Φn> + --- + Cn<Φn|Φn>** - **<Ψ|Ψ> =1** ## Operator - **AΨ = λΦ** - **A** = Operator - **Ψ** = Eigen function - **λ** = Eigen value - **Φ** = New Eigen function ## Types of Operator ### Linear Operator: A - **<Ψ> = <Ψ1> + <Ψ2> + ---- + <Ψn>** - **A<Ψ> = <Ψ1> + <Ψ2> + ---- + <Ψn>** - A - Linear operator ### Property of Linear Operator 1. **A+B = B+A** - Commutative Law 2. **A+B ≠ B+A** 3. **A(B+C) = A.B + A.C** - Associative Law ## Null Vector - **Ô** - **ÔΨ = 0** ## Identity Operator - **1Ψ = Ψ** ## Power Operator - **ÂP = Â.Â.Â.Â--- P times** - **P = A^n** ## Function Operator - **FΨ = a∂²Ψ/∂x² + b∂²Ψ/∂x + ∂Ψ/∂x** ## Inverse Operator - **<Ψ|A + <Ψ|A = <Ψ| + <Ψ|A** - **A.B = 1** - **A and B** - each other **Inverse Operator** ## Adjointive Operator - **<(Bψ)|Φ>dv = <ψ|(AΦ)>dv** - **Ĥ (Hamiltonian Operator) ** -**Adjointive Operator** ## Unitary Operator (Û): - **ÛÛ* = 1** - **Û*Û=1** ## Harmitian Operator: - It is a self Unitary operator - It follows the integration relation if A is Hamilitonian operator. - **A = A+** - If we have two wave function **Ψ, Φ** - **∫(ÂΨ)*Φdv = ∫Ψ(ÃΦ)*dv** - If any operator follows this and that operator is called **Harmitian Operator** - **A = A+** - Harmitian - **A = - A+** - Anti-Harmitian - **∫(ΦÂΦ)dv = dΨ/dx** ## Properties of Harmitian Operator: 1. Eigen values of H.O is always **not real** - Suppose **A** is **Harmitian operator** and it belongs to two Eigen state **Φα, Φα'** - **AΦα= αΦα - (1)** - **AΦα' = α'Φα' - (2)** - From definition of Harmitian operator - **∫Φα*(AΦα')dv = ∫(ÂΦα)*Φα'dv - (3)** - From equation **(1)** and **(2)** -**α'∫Φα*Φα'dv = α∫Φα*Φα'dv** - **(α'-α)∫Φα*Φα'dv = 0** - If **α = α' then** **∫Φα*Φα'dv = 0** - Φ1 and Φ2 both are **⊥** to each other. - Harmitian operator follows **orthogonality** ## If Two Harmitian Operators Commute: - Then there **is also Harmitian operator** - A & B are **Harmitian operator** if they commute. - **A.B-B.A = 0** ## Harmitian Operator - Harmitian operators are **linear** operator, they follow the **commutative law** - **A.B & B.A** - are also **H.O** if they follow: - **A.B-B.A = 0** ## Any Power of Harmitian Operator - Any power of Harmitian operator is always **Harmitian operator** - **(Â^n)+ = 0** ## Every Operator Commutes Itself - **[Â, Â] = 0** ## Linear Operator - Linear operator of linear combination is Harmitian operator. - **A, B, C** - **k1.A + k2.B + k3.C + ---- = 1** ## Each and Every Harmitian Operators Generated Unitary Operator - **A ** - Harmitian operator, Û - Unitary Operator - **Û=e^(iθA)** - **θ** - is any real number ## Simultaneously Eigen Function: - Suppose we have two **A** and **B** are operators - **Ψ** - is eigen function with eigen value **a b** - **A Ψ = aΨ** - **B Ψ = bΨ** - Then **Ψ** is called simultaneously eigen function of operator **A** and **B** belonging to eigen value **a b** respectively. -**Ψ** is **simultaneously eigen function** of **A** and **B** if it follows below condition. - **Â(B̂Ψ) = B̂(ÂΨ) = a(B̂Ψ) = abΨ** and **Â(B̂Ψ) = B̂(ÂΨ) = b(ÂΨ) = abΨ** - **ÂB̂Ψ = B̂ÂΨ** - **(ÂB̂-B̂Â)Ψ = 0** - **[ÂB̂]= 0** - Then they have **common eigen function** ## Parity Operator or Space Inversion Operator - It changes the sign of **space coordinate** when is operator on any coordinate. - **P^2 ψ(x,y,z) = P(Pψ(x,y,z)) = P(ψ(x,-y,-z))** - **P^2 ψ(x,y,z) = ψ(-x,-y,-z)** - **P^2 ψ(x,y,z) = P(Pψ(x,y,-z))** - **P^2 ψ(x,y,z) = ψ(x,y,z)** - Equation **(2)** and eq **(1)** - **x = ±1 then** **Pψ(x,y,z) = ψ(-x,-y,-z)** - **Even Parity**- if sign of coordinate is **not changed** then parity is called **even parity** - **Odd Parity**- if any space coordinate of any space is changed when **P operator** then the parity is called **odd parity** - **Exam:** - **F = x, x^3** - **F = sin θ** ## Properties of Parity Operator: 1. **Eigen value ±1** 2. **Linear Operator** 3. **Harmitian Operator** 4. It is always commutes with **Harmitian operator** 5. It shows the **constant motion** ## Ket and Bra Notation: - **ket - |i>** - initial state - **bra - <i|** - final state - **<i|A|i> = <A>** - expectation value of operator A - Ket and bra are **complex conjugate** of each other. - **<Ψ| = (|Ψ>)*** - ket to bra **changed** - **<Ψ| = (|Ψ>)*** ## Example 1: - **|Ψ> = [3i , 2+i , 4 ]** - **|Φ> = [2 , -i , (2+3i) ]** - find **<Ψ|** and **<Φ|** - Solution: - **<Ψ| = [3i , 2+i , 4 ]* ** - **<Ψ| = [-3i , 2-i , 4 ]** - **<Φ| = [2 , -i , (2+3i)]** - Prove that **<Ψ|Φ>** are orthogonal: - **= (-3i, 2-i, 4) ( 2 , -i , (2+3i) ) = 0** ## Example 2: - **|Ψ> = [5i , 2+i , i ]** - **|Φ> = [ 3i , 8i , -gi]** - Show that **|Ψ>** is normalized - Solution: - **<Ψ |Ψ> = [5i, 2+i, i] * [5i, 2+i, i]** - **= [-5i, 2-i, -i] [5i, 2+i, i]** - **= -5ix5i + 2(2) + ix(-i)** - **= 25 + 4 +1 = 30**** - **Ψ** is not normalized function. - **a^2* <Ψ|Ψ> = 1** - **a^2 × 30 = 1** - **a = 1/√30** - **|Ψ> = 1/√30 [5i , 2+i , i]** ## Example 3: - **|Ψ> = 2|Ψ1> + 1/2|Ψ2> + 1/4 |Ψ3> - (1)** - Now find the value of **t** - Solution: - **|Ψ> = q1|Ψ1> + q2|Ψ2> + q3|Ψ3>** - **Σ a i^2 = 1** - **t^2 + (1/2)^2 + (1/4)^2 = 1** - **t^2 = 13/16** - **t = √13 / 4** ## Heisenberg's Uncertainty Principle - **Heisenberg uncertainty principle** states that product of the uncertainty in determining the position and momentum of the particle is approximately equal to the number of the order ie. - **Δpx.Δx = h** ## Suppose A and B are two operator - They are **Harmitian operator** is uses as they are **simple observable** - **operator value of A = <Ψ|A|Ψ> = <A>** - **operator value of B = <Ψ|B|Ψ> = ** - It the **error** in the measarment in operator **A and B** are as below - **(ΔA)^2 = <A^2> - (<A>)^2 = <A^2>-(<A>)^2>** - **(ΔB)^2 = <B^2> - ()^2 = <B^2>-()^2>** - **(ΔA) = √<>(2>-(<A>)2>** - **(ΔB) = √<>(2>-()2>** - **ΔA.ΔB ≥ 1/2|<[C.A.B]>|** - **ΔA.ΔB ≥ 1/2 |<[C.A.B]>|** - **Linear Harmitian operator A = A - <A>** - **A, B** are **conjugate Harmitian operator** then we can use this method - **(ΔA)^2 = <A^2> - <A>^2 = (A)^2** - **(ΔB)^2 = <B^2> - ^2 = (B)^2** - Let **C** is **linear combination operator** - **C = A + idB** - **<Ψ|C*C|Ψ> ≥ 0** - **((<C*C>)^2 ≥ 0** - **{<(△A)2 + (△B)2- i< [A,B]> } ≥ 0** - **f(λ) = {<(△A)2 + (△B)2- i< [A,B]> } ≥ 0** - **For minimum value of f(λ)** - **df(λ)/dλ = 0** - **0 + 2λ(ΔB)^2-i<[C,A,B]> = 0** - **2λ0(ΔB)^2 -i<[C,A,B]> = 0** - **λ0 = i<[C,A,B]> / 2(ΔB)^2** - **f(λ0) = <(ΔA)^2 - (i<[C,A,B]) / (2(ΔB)^2) (ΔB)^2 - i<[C,A,B]> ** - **= < (ΔA)^2 + (ΔB)^2 - (i<[C,A,B] / (2(ΔB)^2) [C,A,B]> ** - **f(λ0) = <(ΔA)^2 + (ΔB)^2 - (i[C,A,B]) / (2(ΔB)^2) [C,A,B]> ≥ 0** -<(ΔA)^2 + (ΔB)^2 + i[C,A,B]^2 / (2(ΔB)^2) ≥ 0 -<(ΔA)^2 + (ΔB)^2 ≥ -i[C,A,B]^2 / (2(ΔB)^2) -<(ΔA)^2 (ΔB)^2 ≥ -i[C,A,B]^2 / (2(ΔB)^2) - **<(ΔA)^2(ΔB)^2> ≥ 1/2 |<(C.A.B)>|** - **Δx.Δpx ≥ ħ/2 - (i)** - **ΔE.Δt ≥ ħ/2- (ii)** ## Application of Heisenberg's Uncertainty Principle - **Non existence of electron in the nucleus** - **Δx = 5×10^-15 m** - **Δx.Δpx ≥ ħ/2** - **Δpx ≥ ħ/2 Δx** - **Δpx ≥ 1.05×10^-34 / 2×5×10^-15 = 1.05×10^-19 kgm/s** - **E^2 = C^2p^2 + m0^2C^4** - **C^2p^2 ≥ m0^2 C^4** - **Emin = (3×10^8)^2 × (1.05×10^-20)^2** - **Emin = 3×10^8 × 1.05×10^-20** - **Emin = 3.15×10^-12 J = 20 MeV** - It means if electron exist in the nucleus their energy must be at least 20 MeV, experiment on the other hand indicates that electron emitted from the nuclei during beta decay are never greater then 4 MeV. *This shows that electrons do not exist within the nuclei depenetly* - **Ground state energy of Hydrogen atom and ground state** - **Radius Δx = r0** - **Δx.Δpx ≥ ħ/2** - **Δpx ≥ ħ/2r0** - **E = p^2/2m - Ke^2/r** - **E= ħ^2 / 2mr0^2 - ke^2 / r0 ** - For minimum when **dE/dr = 0** - **dE/dr = -ħ^2 / mr0^3 + Ke^2 / r0^2 = 0** - For **r=r0**: **-ħ^2/mr0^3 + Ke^2/r0^2 = 0** - **ħ^2/mr0^3= Ke^2/r0^2** - **r0^3 = ħ^2 / mKe^2** - **r0 = (ħ^2 / mKe^2)^1/3** - **r0 = (1.05×10^-34)^2 / (9.1×10^-31 × 9×10^9 × 1.6×10^-19)^2** - **r0 = 0.529A°** - **Emin = ħ^2 / 2mr0^2 - Ke^2 / r0** - **Emin = ħ^2 / 8mr0^2 (ħ^2 / mKe^2) - Ke^2 × mKe^2 / ħ^2** - **= mKe^2e^4 / 8ħ^2 - mKe^2e^4 / ħ^2** - **= -mKe^2e^4 / 2ħ^2** - **Emin = -1×9.1×10^-31 × (9×10^9)^2 × (1.6×10^-19)^2 / 2 × (1.05×10^-34)^2 × (1.6×10^-19)^2** - **Emin = - 13.6 eV** - **Ground state energy of Harmonic Oscillator** - **E= px^2/2m + (1/2)mw^2x^2** , w^2 = k/m - **Emin = (Δpx)^2 / 2m + (1/2)mw^2(Δx)^2** - **Δx.Δpx ≥ ħ/2** - **Δpx ≥ ħ/2Δx** - **Δpx = ħ/2Δx** - **Emin = ħ^2 / 8m(2Δx)^2 + (1/2)mw^2(Δx)^2** - **= ħ^2 / 8m(Δx)^2 + (1/2)mw^2(Δx)^2** - For minimum **dF/d(Δx) = 0** - at **Δx = Δx0** - **-ħ^2 / 8m(Δx0)^3 + (1/2)mw^2Δx0 = 0** - **mw^2 Δx0 = ħ^2(Δx0)^3 / 4m** - **(Δx)^4 = ħ^2/ 4m^2w^2** - **(Δx)^2 = ħ / 2mw** - **Emin = ħ^2×2mw/ 8mħ + 1/2mw^2ħ/2mw** - **Emin = ħw/4 + ħw/2 = 3ħw/ 4** - **Emin = 1/2 ħw** , E=(n+1/2)ħw - **n=0 , E= ħw/2** - Which is the required value of the minimum energy of the oscillator. - **Natural width of spectral line** - **ΔE.Δt ≥ ħ/2** - **ΔE = ħ/2Δt** - **ΔE = 1.05×10^-34 / 2×10^-8×1.6×10^-19** - **ΔE = 8.33×10^-8 eV** - **Phase and group velocity:** - **ψ1(x,t) = A sin(ω1t-k1x)** - **ψ2(x,t) = A sin(ω2t-k2x)** - **ψ(x,t) = ψ(x,t) + ψ2(x,t)** - **= A sin(ω1t - k1x) + A sin(ω2t-k2x)** - **= 2A sinf((ω1+w2)t / 2 - (k1+k2)x / 2) cosf((ω1-ω2)t / 2- (k1-k2)x / 2)** - **ω1 = ω2 = ω** - **k1 = k2 = k ** - **k1+k2 = k** - **k1-k2 = 0** - **ω1-ω2 = Δω** - **k1-k2 = Δk** - **ψ(x,t) = 2A cos(Δωt/2 - Δkx/2) sin(ωt-kx)** - **ωt - kx = 0** - **ω = kdx/dt** - **dx/dt = ω/k** - **Group velocity** - If **Δωt - Δkx = 0** - **Δωt = Δkx** - **Δω/Δk = dx/dt** - **dx/dt = Δω/Δk** - **phase velocity** ## Commutator Algebra - **[A,B] = -[B,A]** - **[A,B+C] = [A,B] + [A,C]** - **[A+B,C] = [A,C] + [B,C]** - **[A,BC] = [A,B]C + B[A,C]** - **[A,B.C] = [A, B]C + B[A,C] + B[C,Â] ** - **[A.(B,C)] + [B(C,A)] + [C.(A,B)] = 0** ## Some Operator - **x = x** , **px = -iħ∂/∂x** , A = **p^2/2m + V(r)** - **y = y** , **py = -iħ∂/∂y** - **z = z** , **pz = -iħ∂/∂z** - Find **[x,px] = ?** - **[x,px]ψ = (xpx - pxψ)** - **= -iħx∂ψ/∂x + iħ∂(xψ)/∂x** - **= -iħ∂(x ψ)/∂x + iħ∂ (x ψ)/∂x + iħ ψ** - **= [x,px]ψ = iħψ** - **[x,px] = iħ** - **Find the** - **[x,x ] = 0** - **[px,px ] = 0** - **[y,y ] = 0** - **[py,py ] = 0** - **[z,z ] = 0** - **[pz,pz ] = 0** - **[ý,px ] = ?** - **[py,px ] = ?** - **[z,px ] = ?** - **[pz,px ] = ?** - **Solution** - **[ý,py]ψ = [ýpy - pyψ]** - **= [ý(- iħ∂/∂y) - (-iħ ∂/∂y)ψ]** - **= -iħy∂ψ/∂y + iħ ∂(yψ)/∂y** - **= -iħ∂(y ψ)/∂y + iħ∂ (y ψ)/∂y + iħ ψ** - **= [ý,py]ψ = iħ ψ** - **[ý,py] = iħ ** - Similarly - **[z,pz]ψ = iħψ** - **[x,py]ψ = [xpy - pyx]ψ** - **= x(-iħ∂/∂y) + iħ ∂(xψ)/∂y** - **= x(-iħ∂ψ/∂y) + iħ∂(x ψ)/∂y + iħψ ∂x/∂y ** - **= iħ∂ψ/∂x** - Similarly - **[ý,pz]ψ = iħ∂ψ/∂x** - **[z,px]ψ = iħ∂ψ/∂x** ## Communication Relation - **(i) [x,Ĥ] = ?** , **Ĥ = p^2/2m + V(r)** - **Solution:** **[$^2+V(r)]** - **[p, p^2 / 2m] + [p,V(r)]** - **= 1/2m [[p,p^2]x + p [p,V(r)]x]** - **= 1/2m [(p^2p - p^2 p) + p(p V(r) - V(r)p)]** - **= 1/2m (0 + p (p V(r) + V(r)p + iħ∂(V(r)ψ)/∂x))** - **= -iħ/2 ∂( V(r)ψ)/∂x + iħ/2 ∂(V(r)ψ)/∂x** - **= -iħ/2 (∂(V(r)ψ)/∂x - ∂(V(r)ψ)/∂x) + iħ/2 ∂(V(r)ψ)/∂x + iħ/2 ∂(V(r)ψ)/∂x** - **= -iħ/2 ∂(V(r)ψ)/∂x + iħ/2 ∂(V(r)ψ)/∂x - iħ/2 (∂(V(r)ψ)/∂x + iħ/2 ∂(V(r)ψ)/∂x** - **= -iħ∂(V(r)ψ)/∂x + iħ ∂(V(r)ψ)/∂x** - **= -iħ ∂(V(r)ψ)/∂x + iħ ∂(V(r)ψ)/∂x** - **= [x,Ĥ] = - iħ ∂(V(r)ψ)/∂x** - **(2) [Lx,Lx] = ?** - **Solution:** - **[Lx,Lx]ψ = [Lx,Ly^2 + Lz^2,Lx]** - **[Lx,Lx] = [Lx,Ly^2] + [Lx,Lz^2] + [Lz^2,Lx]** - **= [Lx,Ly^2,Lx] + [Lx,Lz^2,Lx]** - **Imporatant** - **Lx = Ly^2 - Lz^2** - **[Lx,Ly] = iħLz , [Lx,Lz] = -iħLy** - **[Ly,Lz] = iħLx , [Lz,Lx] = -iħLy** - **= Ly[Lx,Ly] + Ly[Lx,Ly] + Lz[Lx,Lz] + Lz[Lx,Lz]** - **= Ly(-iħLz) + (-iħLy)Ly + Lz(iħLy) + (iħLy)Lz** - **= Ly(-iħLz) + (-iħLy)Ly + Lz(iħLy) + (iħLy)Lz** - **[Lx,Lx] = 0** - Similarly - **[Ly,Ly] = 0** - **[Lz,Lz] = 0** - **(3) Prove that [Lx,Ly] = iħLz ** - **L = orbital angular momentum operator** - **L = -iħr x p** - **r = xi + yj + zk** - **p = px^i + py^j + pz^k** - **İ = -iħ [[ý ∂/∂z - z ∂/∂y)] + [z ∂/∂x - x ∂/∂z)] + [x ∂/∂y - y ∂/∂x ]k** - **Lx = -iħ [ý ∂/∂z - z ∂/∂y]** - **Ly = -iħ [z ∂/∂x - x ∂/∂z]** - **Lz = -iħ [x ∂/∂y - y ∂/∂x]** - **[Lx,Ly] = -ħ^2 { ∂/∂x( ∂(yψ)/∂z - z ∂( ψ)/∂y) - ∂/∂y ( ∂(zψ)/∂x - x∂( ψ)/∂z) }** - **= -ħ^2 { ∂^2(y ψ)/∂x∂z - ∂^2(zψ)/∂x∂y - ∂^2(zψ)/∂y∂x + ∂^2(xψ)/∂y∂z }** - **= -ħ^2 { ∂^2(y ψ)/∂x∂z - ∂^2(zψ)/∂x∂y - ∂^2(zψ)/∂y∂x + ∂^2(xψ)/∂y∂z } ** - **= -ħ^2 { ∂^2(y ψ)/∂x∂z - ∂^2(zψ)/∂x∂y - ∂^2(zψ)/∂y∂x + ∂^2(xψ)/∂y∂z }** - **= -ħ^2 { ∂^2(y ψ)/∂x∂z - ∂^2(zψ)/∂x∂y + ∂^2(xψ)/∂y∂z - ∂^2(zψ)/∂y∂x }** - **= -ħ^2 { y ∂^2(ψ)/∂x∂z - z ∂^2(ψ)/∂x∂y + x∂^2(ψ)/∂y∂z - z ∂^2(ψ)/∂y∂x }** - **= -ħ^2 { y ∂^2(ψ)/∂x∂z - z ∂^2(ψ)/∂x∂y + x∂^2(ψ)/∂y∂z - z ∂^2(ψ)/∂y∂x }** - **= -ħ^2 { y ∂^2(ψ)/∂x∂z - z ∂^2(ψ)/∂x∂y + x∂^2(ψ)/∂y∂z - z ∂^2(ψ)/∂y∂x }** - **= -ħ^2 { y ∂^2(ψ)/∂x∂z - z ∂^2(ψ)/∂x∂y + x∂^2(ψ)/∂y∂z - z ∂^2(ψ)/∂y∂x }** - **= -ħ^2 { y ∂^2(ψ)/∂x∂z - z ∂^2(ψ)/∂x∂y + x∂^2(ψ)/∂y∂z - z ∂^2(ψ)/∂y∂x }** - **= -ħ^2 { y ∂^2(ψ)/∂x∂z - z ∂^2(ψ)/∂x∂y + x∂^2(ψ)/∂y∂z - z ∂^2(ψ)/∂y∂x }** - **= -ħ^2 { y ∂^2(ψ)/∂x∂z - z ∂^2(ψ)/∂x∂y - x^2 ∂^2(ψ)/∂y∂z + y^2 ∂^2(ψ)/∂y∂z}** - **similarly [Lx,Lz] =-ħ^2 { ∂/∂x(∂(zψ)/∂y - y∂(ψ)/∂z) - ∂/∂z (∂(yψ)/∂x - x∂(ψ)/∂y)}** - **=-ħ^2 { ∂^2(zψ)/∂x∂y -∂^2(yψ)/∂x∂z - ∂^2(yψ)/∂z∂x + ∂^2(xψ)/∂z∂y}** - **= -ħ^{2} { ∂^2(zψ)/∂x∂y - ∂^2(yψ)/∂x∂z + ∂^2(xψ)/∂z∂y - ∂^2(yψ)/∂z∂x}** - **= -ħ^2 { ∂^2(zψ)/∂x∂y - ∂^2(yψ)/∂x∂z + ∂^2(xψ)/∂z∂y - ∂^2(yψ)/∂z∂x}** - **= -ħ^2 { ∂^2(zψ)/∂x∂y - ∂^2(yψ)/∂x∂z + ∂^2(xψ)/∂z∂y - ∂^2(yψ)/∂z∂x}** - **= -ħ^2 { ∂^2(zψ)/∂x∂y - ∂^2(yψ)/∂x∂z - z∂^2(ψ)/∂x∂y + x∂^2(ψ)/∂z∂y}** - **[Lx,Ly] + [Ly,Lx] = -ħ^2 { ∂^2(ψ)/∂x∂y - ∂^2(ψ)/∂x∂z - z∂^2(ψ

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