Particles and Wave functions [5.1]

The energy of a spin system depends on a particle's (or a system's) angular momentum which is quantized.

Quantum state of a particle in motion

But a particle, for example, an electron, which is moving through an electric field has an energy which depends upon position. We could describe such a particle in terms of a set of basis states which consist of all the positions it could be at. Even for just 1-d motion, the set of all $x$ positions is infinite!

So a function of position might be a more appropriate way to describe the quantum state in this particular basis-of-positions. Formally we write: $$\innerp{x}{\psi} \equiv \psi(x).$$

For our two-state systems, $\ket{\psi}=a\ket + +b\ket -$, where $a=\innerp{+}{\psi}$ and $b=\innerp{-}{\psi}$. The sum over probabilities over all possible outcomes of a $z$ analyzer was 1: $$1=a^*a+b^*b.$$ But now, instead of summing the squared projections of the wave state over just two eigenstates, we will need to sum the squared projections onto all of the infinite $\ket x$ positions. We necessarily resort to an integral: $$1=\sum_{\infty}|\innerp{x}{\psi}|^2 \to \int_{-\infty}^{+\infty}\psi^*(x)\psi(x)\,dx.$$

How to write the energy (that is... the Hamiltonian) of a particle in motion? We often divide the energy in two pieces, the "kinetic" energy of motion and the "potential" energy of position. Where the potential could be the spring potential, the gravitational potential, the electrical potential energy... $$\begineq \hat H\psi(x) &= \left(\frac 12 mv_x^2+V(x)\right)\psi(x)\\ &= \left(\frac 1{2m} (mv_x)^2+V(x)\right)\psi(x)\\ &= \left(\frac 1{2m} (\hat p)^2+V(\hat x)\right)\psi(x)\ \endeq$$

It turns out that we can write these operators:

In the position representation of a quantum state, the operators for position and momentum take the form: $$\hat x\equiv x$$ and $$\hat p\equiv -i\hbar\frac d{dx}.$$

Using these representations of the operators, the eigenvalue equation for eigenstates $\ket{\varphi_{E_i}}$ of the Hamiltonian is a differential equation, $$\begineq\hat H\ket{\varphi_{E_i}} &=E_i\ket{\varphi_{E_i}}\\ \left(\frac 1{2m} (-i\hbar\frac d{dx})^2+V(\hat x)\right)\ket{\varphi_{E_i}}&=E_i\ket{\varphi_{E_i}}\\ \endeq $$

The Schrödinger equation for energy eigenstates: $$\left(\frac {-\hbar^2}{2m} \frac {d^2}{dx^2}+V(\hat x)\right)\varphi_{E_i}(x) = E_i\varphi_{E_i}(x)$$