Question

Find the operator for kinetic energy in x-dimension.​

Answer 1

Answer:

For every observable property of a system there is a corresponding quantum mechanical operator. This is often referred to as the Correspondence Principle.

Answer 2

Answer:

The operator for kinetic energy ${\hat_{T_{x}}$, in x-dimension is given by:

${\hat_{T_{x}}}=-\frac{h^{2}}{8\pi ^{2}m} \frac{\partial ^{2}}{\partial x^{2}}$

Explanation:

According to second postulate of the quantum mechanics, for every observable quantity, there is a hermitian mathematical operator in quantum mechanics.

The operator corresponding to position coordinates x, y. and z axes are obtained by simply multiplication the variable itself.

The operator for linear momentum for x coordinate is a differential operator :

${\hat_{p_{x}}}=\frac{h}{2\pi i} (\frac{\partial }{\partial x} )$

The kinetic energy operator ${\hat_{T_{x}}}$  in the x-direction:

We know that, $K.E. =\frac{1}{2}mV_{x}^{2}=\frac{p^{2}_{x}}{2m}$                                ....................(1)

Substitute the value of linear momentum operator in equation (1):

Therefore, kinetic energy operator:

${\hat_{T_{x}}}=\frac{1}{2m}(\frac{h}{2\pi i}\frac{\partial}{\partial x} ) ^{2}$

${\hat_{T_{x}}}=-\frac{h^{2}}{8\pi ^{2}m} \frac{\partial ^{2}}{\partial x^{2}}$

Hence, the kinetic energy operator in x-direction is obtained.