Question

Can someone please explain and show how we go from equation 5 to equation? 6? Thank you!!

Answer 1

Equation 6 is a rearrangement of the Hamiltonian equation (equation 5). To get from equation 5 to equation 6, we can use the following steps:

Substitute the Lagrangian equation (equation 4) into the Hamiltonian equation (equation 5).

Simplify the resulting expression.

Rearrange the terms to obtain equation 6.

Here’s how we can do it:

H=pxx˙−L=iℏ∂L/∂x

Substituting the Lagrangian equation (equation 4) into the Hamiltonian equation (equation 5), we get:

H=pxx˙−(−T−U)=pxx˙+T+U

Simplifying the expression, we get:

H=2mpx2+21mx˙2+kxe−αx

Rearranging the terms, we get:

H=−2mℏ2∂x2∂2+kxe−αx