The energy of a true polyene, considering it as a particle-in-a-box, is to be stated. Concept introduction: The Schrödinger equation is used to find the allowed energy levels for electronic transitions in the quantum mechanics . It is generally expressed as follows. H ⌢ Ψ = E Ψ Where, • H ⌢ is the Hamiltonian operator. • Ψ is the wavefunction. • E is the energy. The energy obtained after applying the operator on wavefunction is known as the eigen value for the wavefunction.

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A two-level system is in a quantum state = α₁₁ + a22, which can be represented by the vector a = {a1, a2}. We are looking for the conditions under which is eigenstate of the operator c., defined below. 1) In the expression c.ỗ, c is a vector with components {Cr, Cy, Cz} ( C; are real numbers) and ỗ is a vector with components {σx, σy,σz} (σ¿ are 2 × 2 matrices). This means that the operator c. can also be represented by a 2 × 2 matrix. Write the matrix of the operator c. knowing that 0 0x = = (₁ }) = ( 5 ) . Oy σz= 0 (6-99) 1 0 1 0 2) In matrix form, the eigenvalue equation is AX AX, where A is the matrix of the operator of interest, X the column matrix representing the eigenvector and the corresponding eigenvalue. Write the eigenvalue equa- tion that needs to be verified for the quantum state to be an eigenvector of the operator c.. = 3) Note that AX = XX ⇒ (A - I)X 0, where I is the identity matrix. (AAI)X=0 is true if and only if det(A - I) = 0. Solve this equation for the operator…

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Question

Chapter 12, Problem 12.39E

Interpretation Introduction

Interpretation:

The energy of a true polyene, considering it as a particle-in-a-box, is to be stated.

Concept introduction:

The Schrödinger equation is used to find the allowed energy levels for electronic transitions in the quantum mechanics. It is generally expressed as follows.

H⌢Ψ=EΨ

Where,

• H⌢is the Hamiltonian operator.

• Ψis the wavefunction.

• Eis the energy.

The energy obtained after applying the operator on wavefunction is known as the eigen value for the wavefunction.

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A two-level system is in a quantum state = α₁₁ + a22, which can be represented by the vector a = {a1, a2}. We are looking for the conditions under which is eigenstate of the operator c., defined below. 1) In the expression c.ỗ, c is a vector with components {Cr, Cy, Cz} ( C; are real numbers) and ỗ is a vector with components {σx, σy,σz} (σ¿ are 2 × 2 matrices). This means that the operator c. can also be represented by a 2 × 2 matrix. Write the matrix of the operator c. knowing that 0 0x = = (₁ }) = ( 5 ) . Oy σz= 0 (6-99) 1 0 1 0 2) In matrix form, the eigenvalue equation is AX AX, where A is the matrix of the operator of interest, X the column matrix representing the eigenvector and the corresponding eigenvalue. Write the eigenvalue equa- tion that needs to be verified for the quantum state to be an eigenvector of the operator c.. = 3) Note that AX = XX ⇒ (A - I)X 0, where I is the identity matrix. (AAI)X=0 is true if and only if det(A - I) = 0. Solve this equation for the operator…

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