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Suppose we have a particle of mass ##m## moving freely in the xy-plane, except for being constrained by hard walls to have ##-L/2 < y < L/2##. Now, the energy eigenstates would be something like
##\psi (x,y) = C \psi_x (x) \psi_y (y) = C e^{-ikx}\cos\left(\frac{n\pi y}{L}\right) ##,
where ##n## would have to be an odd integer, it seems. So, now the total energy ##E## would be a sum from the free-particle motion in the x-direction and particle-in-box motion in y-direction.
Now, suppose that the 2d "corridor" the particle is moving in is not of constant width ##L##, but has some width ##L_1## when ##x < x_1## and other width ##L_2## when ##x > x_2##. Also, between ##x_1## and ##x_2## the width increases linearly from ##L_1## to ##L_2##.
In the region ##x < x_1## the energy eigenfunctions would be like ##\psi (x,y) = C_1 e^{ik_1 x}\cos\left(\frac{n\pi y}{L_1}\right)## and in region ##x > x_2## they would be like ##\psi (x,y) = C_2 e^{ik_2 x}\cos\left(\frac{n\pi y}{L_2}\right)##. If ##L_2 > L_1##, it would seem to be required that ##k_2 > k_1## to make the total energy eigenvalue the same in both regions despite the ground-state energy of the particle-in-box like y-motion being smaller when the width of the corridor is larger.
Questions:
1. If the distance between ##x_1## and ##x_2## is very large compared compared to the difference ##| L_1 - L_2 |##, can the wavefunction between ##x_1## and ##x_2## be approximated by something like
##\psi (x,y) = Ce^{ik(x)x}\cos\left(\frac{n\pi y}{L(x)}\right)##,
where the functions ##L(x)## and ##k(x)## change linearly between ##x = x_1## and ##x = x_2## ?
2. Is this kind of a problem described in any textbook or journal article ?
##\psi (x,y) = C \psi_x (x) \psi_y (y) = C e^{-ikx}\cos\left(\frac{n\pi y}{L}\right) ##,
where ##n## would have to be an odd integer, it seems. So, now the total energy ##E## would be a sum from the free-particle motion in the x-direction and particle-in-box motion in y-direction.
Now, suppose that the 2d "corridor" the particle is moving in is not of constant width ##L##, but has some width ##L_1## when ##x < x_1## and other width ##L_2## when ##x > x_2##. Also, between ##x_1## and ##x_2## the width increases linearly from ##L_1## to ##L_2##.
In the region ##x < x_1## the energy eigenfunctions would be like ##\psi (x,y) = C_1 e^{ik_1 x}\cos\left(\frac{n\pi y}{L_1}\right)## and in region ##x > x_2## they would be like ##\psi (x,y) = C_2 e^{ik_2 x}\cos\left(\frac{n\pi y}{L_2}\right)##. If ##L_2 > L_1##, it would seem to be required that ##k_2 > k_1## to make the total energy eigenvalue the same in both regions despite the ground-state energy of the particle-in-box like y-motion being smaller when the width of the corridor is larger.
Questions:
1. If the distance between ##x_1## and ##x_2## is very large compared compared to the difference ##| L_1 - L_2 |##, can the wavefunction between ##x_1## and ##x_2## be approximated by something like
##\psi (x,y) = Ce^{ik(x)x}\cos\left(\frac{n\pi y}{L(x)}\right)##,
where the functions ##L(x)## and ##k(x)## change linearly between ##x = x_1## and ##x = x_2## ?
2. Is this kind of a problem described in any textbook or journal article ?