# A physical interpretation of Quantum theory

Exploring Quantum Physics – Week 1 Lecture 2

## Born Interpretation of the Schrödinger equation

Born Rule: $|\Psi(x, y, z; t)|^2 dx dy dz$ is the probability of finding the quantum particle (described by $\Psi(\vec{r}, t)$) in the volume $dV = dx dy dz$ at time t.

## Continuity Equation

$\displaystyle \frac{\partial \rho}{\partial t} + \nabla \cdot \vec{\jmath} = 0 \qquad \rho(\vec{r}, t) = \left| \Psi(\vec{r}, t) \right|^2$
Integrate over volume:
$\displaystyle \frac{\partial \rho}{\partial t} + \int_V d^3r \nabla \cdot \vec{jmath} = 0$
Recognizing Gauss’ theorem in the integral, we get:
$\displaystyle \frac{\partial \rho}{\partial t} + \oint_{\partial V} \vec{\jmath} \cdot d \vec{s}= 0$

From Born’s Rule, calculate the probability flux:
$\displaystyle \dot{P}_v = \frac{\partial}{\partial t} \int_V \Psi^*\Psi d^3r = \int_V \left[ \dot{\Psi}^*\Psi + \Psi^*\dot{\Psi} \right] d^3r$
Remember $|\Psi|^2 = \Psi^*\Psi$

Apply to Schrödinger:
$\displaystyle \begin{array}{ll} i \hbar \dot{\Psi} = \hat{H}\Psi \Rightarrow & \dot{\Psi} = -\frac{i}{\hbar} \hat{H}\Psi \\ & \dot{\Psi}^* = \frac{i}{\hbar} \hat{H}\Psi^* \end{array}$
$\displaystyle \Rightarrow \dot{P}_V = \int_V \left[ \frac{i}{\hbar} \left( \hat{H}\Psi^* \right) \Psi - \frac{i}{\hbar} \Psi^* \left( \hat{H} \Psi \right) \right] d^3r$
Expanding the Hamiltonian and canceling the potential gives:
Remember K.E. is an operator: $\hat{k} = \frac{\hat{\vec{p}}^2}{2m} = -\frac{\hbar^2}{2m} \nabla^2$

$\displaystyle \begin{array}{ll} \dot{P}_V &= -\frac{i\hbar}{2m} \int_V \left[ \left( \nabla^2 \Psi^* \right) \Psi - \Psi^* \left( \nabla^2 \Psi \right) \right] d^3r \\ \\ &= -\frac{i\hbar}{2m} \int_V \nabla \cdot \left[ \left( \nabla \Psi^* \right) \Psi - \left( \nabla \Psi \right) \Psi^* \right] d^3r \\ \\ &= - \oint \vec{\jmath} \cdot d\vec{s} \end{array}$

$\vec{\jmath}$ is the probability current $\rho \vec{v}$
$\displaystyle \Rightarrow \vec{\jmath} = \frac{1}{2} \left( \Psi^* \frac{\hat{\vec{p}}}{m}\Psi - \Psi \frac{\hat{\vec{p}}}{m}\Psi^* \right)$

This connects the change in the probability of finding a particle in V with the flux of the probability current flowing through the surface.

## Quantum operators

Classical properties (momentum, location, etc.) become operators acting on the wave function

• momentum: $\hat{\vec{p}} = -i\hbar \nabla$
• location: $\hat{\vec{r}} = \vec{r} \times$ (multiplication operator)
• kinetic energy: $\hat{K} = -\frac{\hbar^2 \nabla^2}{2m}$

E.g. Angular momentum $\vec{L} = \vec{r} \times \vec{p}$ becomes $\hat{\vec{L}} = \vec{r} \times \hat{\vec{p}} = \vec{r} \times - i \hbar \nabla$

## How do you measure an operator?

Consider measuring particle position…
x becomes a probability function F(x)

Notation: $\langle x \rangle = \mathbb{E}(x) = \int dx\; xF(x)$

$\displaystyle \begin{array}{ll} \Rightarrow \langle \vec{r} \rangle &= \int dV \; \vec{r} | \Psi(\vec{r}) |^2 \\ &= \int dV \; \Psi^*(\vec{r})\vec{r} \Psi(\vec{r}) \end{array}$

This generalizes to:
For a property X with operator $\hat{X}$:
$\displaystyle \langle X \rangle = \int dV \; \Psi^*(\vec{r}) \hat{X} \Psi(\vec{r})$

## Time-independent schrödinger

Recall the time-dependent Schrödinger equation:
$\displaystyle i \hbar \partial_t \Psi(\vec{r}, t) = \hat{H}\Psi(\vec{r}, t) = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r}) \right] \Psi(\vec{r}, t)$

Often, $\hat{H}$ is time independent, so we can separate the variables:
$\displaystyle \Psi(\vec{r}, t) = \Psi(\vec{r})e^{-\frac{i}{\hbar}Et}$
$\displaystyle \Rightarrow i \hbar \partial_t \left( \Psi(\vec{r})e^{\frac{-iEt}{\hbar}} \right) = E \Psi(\vec{r})e^{\frac{-iEt}{\hbar}} = (\hat{H} \Psi(\vec{r})) e^{\frac{-iEt}{\hbar}}$
Canceling the exponential gives us the time-independent Schrödinger equation:
$\displaystyle E \Psi(\vec{r}) = (\hat{H} \Psi(\vec{r}))$
Notice that this looks like an eigenvalue equation, where E is the eigen value and $\Psi$ is the eigen vector.

## Hermitian operators

For any operator, $\hat{A}$, acting in a space of functions $\Psi(\vec{r})$, one can define its Hermitian-adjoint operator $\hat{A}^\dagger$ with:
$\displaystyle \int \Psi^*(\vec{r}) \left[ \hat{A} \Psi(\vec{r}) \right] d^3r = \int \left[ \hat{A}^\dagger \Psi(\vec{r}) \right]^* \Psi(\vec{r}) d^3r$
And $\hat{A}$ is “Hermitian” if $\hat{A}^\dagger = \hat{A}$

Eigenvalue properties: $\hat{A} \Psi_a(\vec{r}) = a \Psi_a(\vec{r})$

• eigenvalues (a) are real if $\hat{A}$ is Hermitian
• eigenvectors, $\Psi_a(\vec{r})$, form a basis

## Physical Significance

• Physical observables in quantum mechanics are described by Hermitian (a.k.a. self-adjoint, though there are subtle differences we will ignore) operators, $\hat{A}^\dagger = \hat{A}$
• Eigenvalues of a physical operator determine possible values of the observable that actually can be measured in an experiment.
$\displaystyle \hat{A}\Psi_a(\vec{r}) = a \Psi_a(\vec{r})$
• Eigenvectors form a basis in the sense that a wave-function can be expressed as their linear combination.
$\displaystyle \Psi(\vec{r}) = \sum_a c_a \Psi_a(\vec{r})$

## Super position principle in Q.M.

• If $\Psi_1(\vec{r}, t)$ and $\Psi_2(\vec{r}, t)$ are solutions to the Schrödinger equation,
$\displaystyle i\hbar \frac{\partial \Psi(\vec{r}, t)}{\partial t} = \left[ - \frac{\hbar^2 \nabla^2}{2m} + V(\vec{r}) \right] \Psi(\vec{r}, t)$,
Then $\Psi(\vec{r}, t) = c_1 \Psi_1(\vec{r}, t) + c_2 \Psi_2(\vec{r}, t)$ is also a solution.
• This motivates the notion of a Hilbert space – a linear vector space where quantum states live.
• The wave function, $\Psi(\vec{r}, t)$, is a specific representation of a quantum state (much like coordinates of a vector).
• Dirac notation for “vectors” of quantum states: $\langle \Psi |$ and $| \Psi \rangle$
• For a basis {$|q\rangle$}:
$\displaystyle \sum_q |q\rangle \langle q| = 1 \quad \mathrm{or} \quad \int_q |q\rangle \langle q| = 1$

## How to choose a basis?

• Physical observables in quantum mechanics are associated with linear Hermitian operators.
• For a generic operator, $\hat{A}$, the eigenvalue problem $\hat{A} |a\rangle = a|a\rangle$ defines eigenvectors that form a basis in the Hilbert space.
• $\Psi(a) = \langle a | \Psi \rangle$ is the wave-function in the a-representation.
• Standard choices are:
• coordinate representation: $\Psi(x) = \langle x | \Psi \rangle$
• momentum representation: $\Psi(p) = \langle p | \Psi \rangle$