10.3 The energy-momentum relation and the Schr¨odinger equation for a free particle

10.3 The energy-momentum relation and the Schr¨odinger equation for a free particle

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We will review this subject in chapter 40 but for now we just need the relationship special relativity posits between energy and momentum. Space and time are put together in “Minkowski space”, which is ${\mathbf{R}}^4$ with indefinite inner product

$$((u_0,u_1,u_2,u_3),(v_0,v_1,v_2,v_3))=-u_0{v}_0+u_1{v}_1+u_2{v}_2+u_3{v}_3$$

Energy and momentum are the components of a Minkowski space vector $(p_0=$ $E,p_1,p_2,p_3)$ with norm-squared given by minus the mass-squared:

$$((E,p_1,p_2,p_3),(E,p_1,p_2,p_3))=-E^2+|\mathbf{p}{|}^2=-m^2$$

This is the formula for a choice of space and time units such that the speed of light is 1. Putting in factors of the speed of light c to get the units right one has

$$E^2-|\mathbf{p}{|}^2{c}^2=m^2{c}^4$$

Two special cases of this are:

• For photons, $m=0,$ , and one has the energy momentum relation $E=|\mathbf{p}|c$

• For velocities v small compared to c (and thus momenta |p| small compared to mc), one has

$$E=\sqrt{|\mathbf{p}{|}^2{c}^2+m^2{c}^4}=c\sqrt{|\mathbf{p}{|}^2+m^2{c}^2}\approx \frac{c|\mathbf{p}{|}^2}{2mc}+m{c}^2=\frac{|\mathbf{p}{|}^2}{2m}+m{c}^2$$

In the non-relativistic limit, we use this energy-momentum relation to describe particles with velocities small compared to c, typically dropping the momentum-independent constant term $m{c}^2$

In later chapters we will discuss quantum systems that describe photons, as well as other possible ways of constructing quantum systems for relativistic particles. For now though, we will just consider the non-relativistic case. To describe a quantum non-relativistic particle we choose a Hamiltonian operator $H$ such that its eigenvalues (the energies) will be related to the momentum operator eigenvalues (the momenta) by the classical energy-momentum relation $\begin{array}{r}E=\frac{|\mathbf{p}{|}^2}{2m}\end{array}$ :

$$H=\frac{1}{2m}(P_1^2+P_2^2+P_3^2)=\frac{1}{2m}|\mathbf{P}{|}^2=\frac{-{\hslash }^2}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)$$

The Schr¨odinger equation then becomes:

$$i\hslash \frac{\partial }{\partial t}\psi (\mathbf{q},t)=\frac{-{\hslash }^2}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)\psi (\mathbf{q},t)=\frac{-{\hslash }^2}{2m}{\nabla }^2\psi (\mathbf{q},t)$$

This is an easily solved simple constant coeficient second-order partial diferential equation. One method of solution is to separate out the time-dependence, by first finding solutions ${\psi }_E$ to the time-independent equation

$$H{\psi }_E(\mathbf{q})=\frac{-{\hslash }^2}{2m}{\nabla }^2{\psi }_E(\mathbf{q})=E{\psi }_E(\mathbf{q})\text{\hspace{1em}(10.3)}$$

with eigenvalue E for the Hamiltonian operator. Then

$$\psi (\mathbf{q},t)={\psi }_E(\mathbf{q})e^{-\frac{i}{\hslash }tE}$$

will give solutions to the full time-dependent equation

$$i\hslash \frac{\partial }{\partial t}\psi (\mathbf{q},t)=H\psi (\mathbf{q},t)$$

The solutions ${\psi }_E(\mathbf{q})$ to the time-independent equation 10.3 are complex exponentials proportional to

$$e^{i(k_1{q}_1+k_2{q}_2+k_3{q}_3)}=e^{i\mathbf{k}\cdot \mathbf{q}}$$

satisfying

$$\frac{-{\hslash }^2}{2m}{i}^2|\mathbf{k}{|}^2=\frac{{\hslash }^2|\mathbf{k}{|}^2}{2m}=E$$

We have thus found that solutions to the Schr¨odinger equation are given by linear combinations of states |k⟩ labeled by a vector $k$, which are eigenstates of the momentum and Hamiltonian operators with

$$P_j|\mathbf{k}⟩=\hslash k_j|\mathbf{k}⟩,H|\mathbf{k}⟩=\frac{{\hslash }^2}{2m}|\mathbf{k}{|}^2|\mathbf{k}⟩$$

These are states with well-defined momentum and energy

$$p_j=\hslash k_j,E=\frac{|\mathbf{p}{|}^2}{2m}$$

so they satisfy exactly the same energy-momentum relations as those for a classical non-relativistic particle.

While the quantum mechanical state space $H$ contains states with the classical energy-momentum relation, it also contains much, much more since it includes linear combinations of such states. At $t$ = 0 the state can be a sum

$$|\psi ⟩=\sum _{\mathbf{k}}{c}_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{q}}$$

where $c_{\mathbf{k}}$ are complex numbers. This state will in general not have a welldefined momentum, but measurement theory says that an apparatus measuring the momentum will observe value ℏk with probability

$$\frac{|c_{\mathbf{k}}{|}^2}{{\sum }_{{\mathbf{k}}^{\prime }}|c_{{\mathbf{k}}^{\prime }}{|}^2}$$

The time-dependent state will be

$$|\psi (t)⟩=\sum _{\mathbf{k}}{c}_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{q}}{e}^{-it\hslash \frac{|\mathbf{k}{|}^2}{2m}}$$

Since each momentum eigenstate evolves in time by the phase factor

$$e^{-it\hslash \frac{|\mathbf{k}{|}^2}{2m}}$$

the probabilities of observing a momentum value stay constant in time.

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