Relation Between the Wave Function and Probability Density

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

The quantity $|\Psi ({\textbf {r}},t)|^{2}$ can be interpreted as probability density. In order for us to do so, two conditions must be met. First, the probability amplitude must be positive semi-definite (equal to or greater than zero). This condition is trivial because $|\Psi ({\textbf {r}},t)|^{2}\!$ is always a non-negative function. Second, the probability density, integrated over all space, must be equal to one:

$\int _{-\infty }^{\infty }d^{3}{\textbf {r}}\,|\Psi ({\textbf {r}},t)|^{2}=1$

We will show that, if this relation is satisfied for a specific time, then it is satisfied for all times shortly.

Because of the fact that we may interpret $|\Psi ({\textbf {r}},t)|^{2}$ as a probability density, we may calculate expectation values of observables, such as position and momentum, in terms of it. In general, the expectation value of an observable $Q({\textbf {r}},{\textbf {p}};t)$ is given by

$\langle Q({\textbf {r}},{\textbf {p}};t)\rangle =\int _{-\infty }^{\infty }d^{3}{\textbf {r}}\,\Psi ^{\ast }({\textbf {r}},t)Q\left({\textbf {r}},-i\hbar \nabla ;t\right)\Psi ({\textbf {r}},t).$

In particular, the expectation value of a position coordinate $x_{i}$ is

$\langle x_{i}\rangle =\int _{-\infty }^{\infty }d^{3}{\textbf {r}}\,x_{i}|\Psi ({\textbf {r}},t)|^{2}$

and that for a component of momentum $p_{i}$ is

$\langle p_{i}\rangle =\int _{-\infty }^{\infty }d^{3}{\textbf {r}}\,\Psi ^{\ast }({\textbf {r}},t)\left(-i\hbar {\frac {\partial }{\partial x_{i}}}\right)\Psi ({\textbf {r}},t).$

Conservation of Probability

We will now show that the solution to the Schrödinger equation conserves probability, i.e. the probability to find the particle somewhere in the space does not change with time. To see that it does, consider

$i\hbar {\frac {\partial }{\partial t}}\Psi ({\textbf {r}},t)=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\textbf {r}})\right]\Psi ({\textbf {r}},t)$

Now multiply both sides by the complex conjugate of $\psi ({\textbf {r}},t):\!$

$i\hbar \Psi ^{\ast }({\textbf {r}},t){\frac {\partial }{\partial t}}\Psi ({\textbf {r}},t)=\Psi ^{\ast }({\textbf {r}},t)\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\textbf {r}})\right]\Psi ({\textbf {r}},t)$

Now, take the complex conjugate of this entire expression:

$-i\hbar \Psi ({\textbf {r}},t){\frac {\partial }{\partial t}}\Psi ^{\ast }({\textbf {r}},t)=\Psi ({\textbf {r}},t)\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\textbf {r}})\right]\Psi ^{\ast }({\textbf {r}},t)$

Taking the difference of the above equations, we finally find

${\frac {\partial }{\partial t}}\left(\Psi ^{\ast }({\textbf {r}},t)\Psi ({\textbf {r}},t)\right)+{\frac {\hbar }{2im}}\nabla \cdot \left[\Psi ^{\ast }({\textbf {r}},t)\nabla \Psi ({\textbf {r}},t)-(\nabla \Psi ^{\ast }({\textbf {r}},t))\Psi ({\textbf {r}},t)\right]=0$

Note that this is in the form of a continuity equation,

${\frac {\partial }{\partial t}}\rho ({\textbf {r}},t)+\nabla \cdot {\textbf {j}}({\textbf {r}},t)=0,$

where

$\rho ({\textbf {r}},t)=\Psi ^{\ast }({\textbf {r}},t)\Psi ({\textbf {r}},t)\!$

is the probability density, and

${\textbf {j}}({\textbf {r}},t)=-{\frac {i\hbar }{2m}}\left[\Psi ^{\ast }({\textbf {r}},t)\nabla \Psi ({\textbf {r}},t)-(\nabla \Psi ^{\ast }({\textbf {r}},t))\Psi ({\textbf {r}},t)\right]$

is the probability current.

Once we know that the densities and currents constructed from the solution of the Schrödinger equation satisfy the continuity equation, it is easy to show that the probability is conserved.

To see this, note that

${\frac {d}{dt}}\int d^{3}{\textbf {r}}|\Psi ({\textbf {r}},t)|^{2}=-\int d^{3}{\textbf {r}}(\nabla \cdot {\textbf {j}})=-\oint d{\textbf {A}}\cdot {\textbf {j}}=0.$

Here, we used the fact that the wave function is assumed to vanish outside of the boundary, and thus the current vanishes as well. Therefore, we see that the normalization of the wave function does not change over time, so that we only need to normalize it at one instant in time, as asserted earlier.

Problems

(1) Consider a particle moving in a potential field $V({\textbf {r}}).$

(a) Prove that the average energy is $\langle E\rangle =\int W\,d^{3}{\textbf {r}}=\int \left({\frac {\hbar ^{2}}{2m}}\nabla \Psi ^{\ast }\cdot \nabla \Psi +\Psi ^{\ast }V\Psi \right)\,d^{3}{\textbf {r}},$ where $W$ is energy density.

(b) Prove the energy conservation equation, ${\frac {\partial W}{\partial t}}+\nabla \cdot {\textbf {S}}=0,$ where ${\textbf {S}}=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial \Psi ^{\ast }}{\partial t}}\nabla \Psi +{\frac {\partial \Psi }{\partial t}}\nabla \Psi ^{\ast }\right)$ is the energy flux density.

Solution

(2) Assume that the Hamiltonian for a system of $N$ particles is ${\hat {H}}=-\sum _{i=1}^{N}{\frac {\hbar }{2m}}\nabla _{i}^{2}+\sum _{i=1}^{N}\rho _{ij}(|{\textbf {r}}_{i}-{\textbf {r}}_{j}|)$, and $\Psi ({\textbf {r}}_{1},{\textbf {r}}_{2},\ldots ,{\textbf {r}}_{N};t)$ is the wave fuction. Defining

$\rho ({\textbf {r}},t)=\sum \rho _{i}({\textbf {r}},t),$

${\textbf {j}}({\textbf {r}},t)=\sum {\textbf {j}}_{i}({\textbf {r}},t),$

$\rho _{1}({\textbf {r}}_{1},t)=\int \cdots \int d^{3}{\textbf {r}}_{2}\,d^{3}{\textbf {r}}_{3}\,\cdots \,d^{3}{\textbf {r}}_{N}\,\Psi ^{\star }\Psi ,$

${\textbf {j}}_{1}({\textbf {r}}_{1},t)=-{\frac {i\hbar }{2m}}\int \cdots \int d^{3}{\textbf {r}}_{2}\,d^{3}{\textbf {r}}_{3}\,\cdots \,d^{3}{\textbf {r}}_{N}\,(\Psi ^{\star }\nabla _{1}\Psi -\Psi \nabla _{1}\Psi ^{\star }),$

and similarly for the other $\rho _{i}$ and ${\textbf {j}}_{i}$, prove the following relation:

${\frac {\partial \rho }{\partial t}}+\nabla \cdot {\textbf {j}}=0$

Solution