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
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