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A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with absolute certainty, the state is represented by the ket R) (or |L)). The most general state vector can then be written as: la)= |R) (Ra) + L) (La) where (Ra) and (La) can be regarded as 'wave-functions in this basis. The particle can tunnel through the partition, this tunneling process is characterized by a Hamiltonian: H = A(L) (R| + |R){L|) where A is a number with dimensions of energy! (i) Find the normalized energy eigenvectors. What are the corresponding energy eigen- values? (ii) In the Schrödinger picture, the basis kets |R) and L) are fixed, and the state vector moves in time. Suppose the system is represented by the ket la) as given above at t = 0. Find the state vector la, to = 0; t) for t> 0 by applying the appropriate time evolution operator to la). (iii) Suppose that at t=0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (iv) Write down the coupled Schrödinger equations for the wave functions (Rla, to = 0; t) and (Lla, to = 0; t). Show that the solutions to the coupled equations are what you expect from (ii). (v) Suppose the printer made an error and wrote H as: H = A|L) (R| By explicitly solving the most general time-evolution problem with this Hamiltonian, show that the probability conservation is violated. Why?

A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with absolute certainty, the state is represented by the ket R) (or |L)). The most general state vector can then be written as: la)= |R) (Ra) + L) (La) where (Ra) and (La) can be regarded as 'wave-functions in this basis. The particle can tunnel through the partition, this tunneling process is characterized by a Hamiltonian: H = A(L) (R| + |R){L|) where A is a number with dimensions of energy! (i) Find the normalized energy eigenvectors. What are the corresponding energy eigen- values? (ii) In the Schrödinger picture, the basis kets |R) and L) are fixed, and the state vector moves in time. Suppose the system is represented by the ket la) as given above at t = 0. Find the state vector la, to = 0; t) for t> 0 by applying the appropriate time evolution operator to la). (iii) Suppose that at t=0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (iv) Write down the coupled Schrödinger equations for the wave functions (Rla, to = 0; t) and (Lla, to = 0; t). Show that the solutions to the coupled equations are what you expect from (ii). (v) Suppose the printer made an error and wrote H as: H = A|L) (R| By explicitly solving the most general time-evolution problem with this Hamiltonian, show that the probability conservation is violated. Why?

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A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with absolute certainty, the state is represented by the ket |R) (or |L)). The most general state vector can then be written as: la) = |R) (Ra) + L) (La) where (Ra) and (La) can be regarded as 'wave-functions in this basis. The particle can tunnel through the partition, this tunneling process is characterized by a Hamiltonian: H = A(L) (R| + |R) (L) where A is a number with dimensions of energy. (i) Find the normalized energy eigenvectors. What are the corresponding energy eigen- values? (ii) In the Schrödinger picture, the basis kets |R) and L) are fixed, and the state vector moves in time. Suppose the system is represented by the ket la) as given above at t = 0. Find the state vector la, to = 0; t) for t> 0 by applying the appropriate time evolution operator to la). (iii) Suppose that at t=0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (iv) Write down the coupled Schrödinger equations for the wave functions (Rla, to = 0; t) and (Lla, to = 0; t). Show that the solutions to the coupled equations are what you expect from (ii). (v) Suppose the printer made an error and wrote H as: H = A|L) (R| By explicitly solving the most general time-evolution problem with this Hamiltonian, show that the probability conservation is violated. Why?
Transcribed Image Text:A box containing a particle is divided into a right and a left compartment by a thin partition. If the particle is known to be on the right (left) side with absolute certainty, the state is represented by the ket |R) (or |L)). The most general state vector can then be written as: la) = |R) (Ra) + L) (La) where (Ra) and (La) can be regarded as 'wave-functions in this basis. The particle can tunnel through the partition, this tunneling process is characterized by a Hamiltonian: H = A(L) (R| + |R) (L) where A is a number with dimensions of energy. (i) Find the normalized energy eigenvectors. What are the corresponding energy eigen- values? (ii) In the Schrödinger picture, the basis kets |R) and L) are fixed, and the state vector moves in time. Suppose the system is represented by the ket la) as given above at t = 0. Find the state vector la, to = 0; t) for t> 0 by applying the appropriate time evolution operator to la). (iii) Suppose that at t=0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (iv) Write down the coupled Schrödinger equations for the wave functions (Rla, to = 0; t) and (Lla, to = 0; t). Show that the solutions to the coupled equations are what you expect from (ii). (v) Suppose the printer made an error and wrote H as: H = A|L) (R| By explicitly solving the most general time-evolution problem with this Hamiltonian, show that the probability conservation is violated. Why?
स्पाग मंग 5a2 107 = (6) 15 = (1)
Transcribed Image Text:स्पाग मंग 5a2 107 = (6) 15 = (1)
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