Advanced Solid State Physics

3.1 Quantization

For the quantization of a given system withwell defined equations of motionit is necessary to
guess a LagrangianL(qi,q ̇i), i.e. to construct a Lagrangian by inspection. qiare the positional
coordinates andq ̇ithe generalized velocities.

With theEuler-Lagrange equations

d

dt

∂L

∂q ̇i

−

∂L

∂qi

= 0 (3)

it must be possible to come back to the equations of motion with the constructed Lagrangian.^1

The next step is toget the conjugate variablepi:

pi=

∂L

∂q ̇i

(4)

Then theHamiltonianhas to be derived with a Legendre transformation:

H=

∑

i

piq ̇i−L (5)

Now theconjugate variables have to be replaced by the given operators in quantum
mechanics. For example the momentumpwith

p→−i~∇.

Now the Schrödinger equation

HΨ(q) =EΨ(q) (6)

is ready to be evaluated.

(^1) There is no standard procedure for constructing a Lagrangian like in classical mechanics where the Lagrangian is
constructed with the kinetic energyTand the potentialU:
L=T−U=^12 mx ̇^2 −U
The massmis the big problem - it isn’t always as well defined as for example in classical mechanics.