Manish Jain

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Let's investigate a three state quantum system (|1?, |2?, |3?). The Hamiltonian for this system is given by: H ? [E1 0 A; 0 E0 0; A 0 E1] In the |1? ? [1; 0; 0], |2? ? [0; 1; 0], |3? ? [0; 0; 1] basis the Hamiltonian is NOT diagonal. 1. Are the state vectors, |1?, |2?, |3?, energy eigenstates? How can you tell? 2. Diagonalize H and find the energy eigenvalues. You should find three distinct values (E1 - ?, E0, and E1 + ?). What is the value of question mark? 3. Sketch an energy level diagram for this system. You can assume E0 < E1 and A < (E1 - E0). What is the ground state, the first excited state, the second excited state? How much energy would be needed to make the transition between the ground state and the two different excited states? 4. Now that you have found the energy eigenvalues, use those eigenvalues to determine the energy eigenstates in terms of the |1?, |2?, |3? basis. Which eigenstate corresponds to the ground state? The first excited state? The second excited state?

          Let's investigate a three state quantum system (|1?, |2?, |3?). The Hamiltonian for this system is given by:
H ? [E1 0 A; 0 E0 0; A 0 E1]
In the |1? ? [1; 0; 0], |2? ? [0; 1; 0], |3? ? [0; 0; 1] basis the Hamiltonian is NOT diagonal.
1. Are the state vectors, |1?, |2?, |3?, energy eigenstates? How can you tell?
2. Diagonalize H and find the energy eigenvalues. You should find three distinct values (E1 - ?, E0, and E1 + ?). What is the value of question mark?
3. Sketch an energy level diagram for this system. You can assume E0 < E1 and A < (E1 - E0). What is the ground state, the first excited state, the second excited state?
How much energy would be needed to make the transition between the ground state and the two different excited states?
4. Now that you have found the energy eigenvalues, use those eigenvalues to determine the energy eigenstates in terms of the |1?, |2?, |3? basis. Which eigenstate corresponds to the ground state? The first excited state? The second excited state?

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Let's investigate a three state quantum system (|1?, |2?, |3?). The Hamiltonian for this system is given by:
H ? [E1 0 A; 0 E0 0; A 0 E1]
In the |1? ? [1; 0; 0], |2? ? [0; 1; 0], |3? ? [0; 0; 1] basis the Hamiltonian is NOT diagonal.
1. Are the state vectors, |1?, |2?, |3?, energy eigenstates? How can you tell?
2. Diagonalize H and find the energy eigenvalues. You should find three distinct values (E1 - ?, E0, and E1 + ?). What is the value of question mark?
3. Sketch an energy level diagram for this system. You can assume E0 < E1 and A < (E1 - E0). What is the ground state, the first excited state, the second excited state?
How much energy would be needed to make the transition between the ground state and the two different excited states?
4. Now that you have found the energy eigenvalues, use those eigenvalues to determine the energy eigenstates in terms of the |1?, |2?, |3? basis. Which eigenstate corresponds to the ground state? The first excited state? The second excited state?

Added by Michael S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Manish Jain

12/19/2021

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The Hamiltonian is the operator that corresponds to the total energy of the system, and its eigenvalues correspond to the possible energy levels of the system. If the Hamiltonian is not diagonal in the |1), |2), |3) basis, this means that these states are not Show more…

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Let's investigate a three state quantum system (|1⟩, |2⟩, |3⟩). The Hamiltonian for this system is given by: H ≐ [E1 0 A; 0 E0 0; A 0 E1] In the |1⟩ ≐ [1; 0; 0], |2⟩ ≐ [0; 1; 0], |3⟩ ≐ [0; 0; 1] basis the Hamiltonian is NOT diagonal. 1. Are the state vectors, |1⟩, |2⟩, |3⟩, energy eigenstates? How can you tell? 2. Diagonalize H and find the energy eigenvalues. You should find three distinct values (E1 - ?, E0, and E1 + ?). What is the value of question mark? 3. Sketch an energy level diagram for this system. You can assume E0 < E1 and A < (E1 - E0). What is the ground state, the first excited state, the second excited state? How much energy would be needed to make the transition between the ground state and the two different excited states? 4. Now that you have found the energy eigenvalues, use those eigenvalues to determine the energy eigenstates in terms of the |1⟩, |2⟩, |3⟩ basis. Which eigenstate corresponds to the ground state? The first excited state? The second excited state?

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Transcript

00:01 In this question we have given a pv diagram of the process that is a b c and we have to find the temperature of the gas at point a b and c so first of all i'm doing the given pv diagram for this tuition so this is our pressure axis and this is volume axis and this is the process a b and this is the process from b to c so this is a this is b and this is c we have given the volume at point a and that is 0 .01 and at b the volume is this is 0 .07 meter cube and the pressure at this point is 2 atmospheric and this point the pressure is 8 so this is our pressure axis this is our volume axis so first of all we have given the number of moles that is n is equal to 0 .45 moles now we have to find the temperature at point a b and c so we can say that from ideal gas equation or from gas equation that is pv is equals to nrt.

01:16 Now from here we get the value of temperature so this will come out to be pv divides by nr.

01:23 So in the first part of the problem the temperature at state a is given by that is pressure at a into volume at a upon n into r.

01:34 Now put all the data in this equation.

01:36 So this is 2 into 10 to the temperature.

01:38 Power 5 into this is 0 .01 upon 0 .45 into this is 8 .315 after solving this we get this is 535 kelvin so this is the temperature at a similarly we can find the temperature at state b and this is given by p b into vb upon this is n into r now put all the data in this equation so pressure at this is 5 into 10 to the power 5 into this is 0 .07 upon 0 .45 into 8 .315 and after solving this we get this is 9 350 kelvin.

02:18 Now temperature at c is given by this is pressure at c into volume at c upon this is n into r so this is 8 into 10 to the power 5 into 0 .07 upon 0 .45 into this is 8 .35 into this is 8 .315 and this will come out to be 15 ,000 kelvin.

02:40 So this is the temperature at state c.

02:43 This is temperature at b and this is temperature at a.

02:52 Now in the second part of the problem, we have to find the work done by or on the gas in this process...

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