Question

Show by direct substitution that eq. 5.39 gives a solution to the two-dimensional Schrödinger equation, eq. 5.37. Find the relationship between kx, ky, and e.

a) √(kx² + ky²) = e
b) kx + ky = e
c) kx² + ky² = e
d) e² = kx² + ky²

Answer 1

To show that eq. 5.39 gives a solution to the two-dimensional Schrödinger equation, we need to substitute the given wave function into the equation and see if it satisfies it. By expanding the differential operators and simplifying, we can show that the left side of the equation is equal to k²Y(x, y). Therefore, the relationship between kx, ky, and e is: √(kx² + ky²) = e.

Step-by-step explanation:

To show that eq. 5.39 gives a solution to the two-dimensional Schrödinger equation, eq. 5.37, we need to substitute the given wave function into the equation and see if it satisfies it. The two-dimensional Schrödinger equation is given by:

[(-ħ²/2m)(∂²/∂x² + ∂²/∂y²) + U(x, y)]Y(x, y) = EY(x, y)

Let's substitute the given wave function Y(x, y) = Ae^(i(kx - wt)) into the equation:

[(-ħ²/2m)(∂²/∂x² + ∂²/∂y²) + U(x, y)](Ae^(i(kx - wt))) = E(Ae^(i(kx - wt)))

By expanding the differential operators and simplifying, we can show that the left side of the equation is equal to k²Y(x, y):

k²Y(x, y) + U(x, y)Y(x, y) = EY(x, y)

This matches with eq. 5.39, which shows that Y(x, y) = Ae^(i(kx - wt)) is indeed a solution to the two-dimensional Schrödinger equation.

Therefore, the relationship between kx, ky, and e is: √(kx² + ky²) = e