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 certainty, the state is represented by the position eigenket |R) (or |L)). Here, we neglect spatial variations within each half box. The most general state vector can be written as la) |R)(Ra) |LX(L\a). (1) where the complex numbers (R\a) and (Lla) can be regarded as "wavefunctions" for the system The particle can tunnel through the partition; this tunneling effect can be characterized by the Hamiltonian A(L){R] + |R){L|), НЕ (2) where Ais a real number with dimensions of energy a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues? b. In the Schrodinger picture, the kets |R) and |L) are fixed, and the state vector moves with time. Suppose the system is represented by a) as given about at t = 0. Find the state vector |a, to = 0; t) for a later time t by applying the appropriate time-evolution to la) c. Suppose at t observing the the particle on the left hand side at a later time t? = 0 the particle is on the right side with certainty. What is the probability for d. Write down the coupled Schrödinger equations for the wavefunctions (R|a, to 0; t). Show that the solutions to the coupled Schrödinger equations are just what you expect from part (b) 0; t) and (Lla, to e. Suppose that the printer made an error, and wrote H as H A|L)(R\. (3) By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated
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 certainty, the state is represented by the position eigenket |R) (or |L)). Here, we neglect spatial variations within each half box. The most general state vector can be written as la) |R)(Ra) |LX(L\a). (1) where the complex numbers (R\a) and (Lla) can be regarded as "wavefunctions" for the system The particle can tunnel through the partition; this tunneling effect can be characterized by the Hamiltonian A(L){R] + |R){L|), НЕ (2) where Ais a real number with dimensions of energy a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues? b. In the Schrodinger picture, the kets |R) and |L) are fixed, and the state vector moves with time. Suppose the system is represented by a) as given about at t = 0. Find the state vector |a, to = 0; t) for a later time t by applying the appropriate time-evolution to la) c. Suppose at t observing the the particle on the left hand side at a later time t? = 0 the particle is on the right side with certainty. What is the probability for d. Write down the coupled Schrödinger equations for the wavefunctions (R|a, to 0; t). Show that the solutions to the coupled Schrödinger equations are just what you expect from part (b) 0; t) and (Lla, to e. Suppose that the printer made an error, and wrote H as H A|L)(R\. (3) By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated
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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 certainty, the state is represented by the position
eigenket |R) (or |L)). Here, we neglect spatial variations within each half box. The most general
state vector can be written as
la) |R)(Ra) |LX(L\a).
(1)
where the complex numbers (R\a) and (Lla) can be regarded as "wavefunctions" for the system
The particle can tunnel through the partition; this tunneling effect can be characterized by the
Hamiltonian
A(L){R] + |R){L|),
НЕ
(2)
where Ais a real number with dimensions of energy
a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues?
b. In the Schrodinger picture, the kets |R) and |L) are fixed, and the state vector moves with time.
Suppose the system is represented by a) as given about at t = 0. Find the state vector |a, to = 0; t)
for a later time t by applying the appropriate time-evolution to la)
c. Suppose at t
observing the the particle on the left hand side at a later time t?
= 0 the particle is on the right side with certainty. What is the probability for
d. Write down the coupled Schrödinger equations for the wavefunctions (R|a, to
0; t). Show that the solutions to the coupled Schrödinger equations are just what you expect from
part (b)
0; t) and (Lla, to
e. Suppose that the printer made an error, and wrote H as
H A|L)(R\.
(3)
By explicitly solving the most general time-evolution problem with this Hamiltonian, show that
probability conservation is violated](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9faf22f-10a8-4cb5-bec9-d63452f6293a%2F4ca3a061-e2d3-408d-b364-015701542586%2F0ucujty.png&w=3840&q=75)
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 certainty, the state is represented by the position
eigenket |R) (or |L)). Here, we neglect spatial variations within each half box. The most general
state vector can be written as
la) |R)(Ra) |LX(L\a).
(1)
where the complex numbers (R\a) and (Lla) can be regarded as "wavefunctions" for the system
The particle can tunnel through the partition; this tunneling effect can be characterized by the
Hamiltonian
A(L){R] + |R){L|),
НЕ
(2)
where Ais a real number with dimensions of energy
a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues?
b. In the Schrodinger picture, the kets |R) and |L) are fixed, and the state vector moves with time.
Suppose the system is represented by a) as given about at t = 0. Find the state vector |a, to = 0; t)
for a later time t by applying the appropriate time-evolution to la)
c. Suppose at t
observing the the particle on the left hand side at a later time t?
= 0 the particle is on the right side with certainty. What is the probability for
d. Write down the coupled Schrödinger equations for the wavefunctions (R|a, to
0; t). Show that the solutions to the coupled Schrödinger equations are just what you expect from
part (b)
0; t) and (Lla, to
e. Suppose that the printer made an error, and wrote H as
H A|L)(R\.
(3)
By explicitly solving the most general time-evolution problem with this Hamiltonian, show that
probability conservation is violated
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