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r7 - 18 Jul 2007 - 23:35:03 - ThanhTruongYou are here: TWiki >  Main Web > LectureNotes > UndergraduatePChem > Chapter3

The Quantum Mechanical Postulates

Quantum Mechanics (QM) is based on a small number of Postulates.  These postulates form the basis for all developments to explain the physical world around us.

Postulate 1:  Description on the State of the QM system

The state of a QM system is completely specified by a wavefunction, $\Psi$, which is a continuous, finite, and singled-valued function of time and of the coordinates of the particles of the system.  The probability of finding the particle in a small volume dv around the point ${\vec r}_{0}$ at time ${t}_{0}$ is $ \left[ { \Psi }^{ *} \left ( { \vec r}_{ 0}, { t}_{ 0} \right ) \Psi \left ( { \vec r}_{ 0}, { t}_{ 0} \right ) dv \right ]$.

Postulate 2:  Time Evolution of the State of the QM system 

The time evolution of a QM system is governed by the time-dependent Schrödinger Equation:

\[\hat H \Psi = i \hbar \frac{ \partial \Psi }{ \partial t}\]

where $\hat H$ is the Hamiltonian operator.

Postulate 3:  Hermitian operator corresponding to every mechanical (measurable) property

For every measurable property of the system in classical mechanics, there exists an one-to-one corresponding Hermitian operator in quantum mechanics.

Observable Operator  Symbol Operator
Total Energy $\hat H$ $-\frac{\hbar^2}{2m} \frac{ { d}^{ 2} }{ d { x}^{ 2} } + V(x)$  
Position   $\hat x$   $x$
Momentum $\hat {p}_{x}$   $\frac {\hbar}{i} \frac {d}{dx}$ 
Kinetic Energy  $\hat T$  $-\frac {{\hbar}^{2}}{2m}{\nabla}^{2}$ 
Potential Energy  $\hat V$  V(x) 

Postulate 4:  Result of a Measurment 

If an observable corresponding to a QM operator $\hat A$ is to be measured without experimental error, the only possible values to be observed are the eigenvalues of the operator $\hat A$.

Postulate 5:  Expectation Value

If the QM system is in the state described by the wavefunction $\Psi$ immediately prior to the measurement of a mechanical variable A on a number of idential replica, the expectation value (the average value) of these measurements is given by

\[<A> = \frac{ \int { \Psi }^{ *} \hat A \Psi d \tau }{ \int { \Psi }^{ *} \Psi d \tau }\]

Consequences of Operator Commutation in QM

Operators that commute can share a set of eigenfunctions and thus the physical properties corresponding to these operators can be measured simultaneously to any precision and vice versa.

Examples:

$\left[\hat V(x), \hat x \right] = V(x) x - xV(x) = 0$, and thus the location and the potential energy of the particle can be measured simultaneously and precisely. On the other hand,

\[ \left[ \hat { P}_{ x}, \hat x \right ] = \hat { P}_{ x}\hat x - \hat x \hat { P}_{ x}\]

From the first term,

\[\hat { P}_{ x}\hat x \psi (x) = \left ( -i \hbar \frac{ d}{ dx}\right ) x \psi (x) = -i \hbar \left[ \psi (x) + x \frac{ d \psi (x)}{ dx} \right ] \]

and the second term,

\[\hat x \hat { P}_{ x} \psi (x) = x \left ( -i \hbar \frac{ d}{ dx}\right )\psi (x) = -i \hbar \left[ x \frac{ d \psi (x)}{ dx} \right ]\]

Thus,

\[ \left ( \hat { P}_{ x} \hat x - \hat x \hat { P}_{ x}\right ) \psi (x) = -i \hbar \psi (x)\]

or

\[\left[ \hat { P}_{ x}, \hat x \right ] = -i \hbar \]

Hence, we cannot measure the momentum and position of the particle simultaneously to an arbitrary precision. This is in fact the Uncertainty Principle that we discussed earlier!

In fact, if we use the standard deviation ${\sigma}_{a}$ as the quantitative statistical measures of the uncertainty in the observed values of the physical property corresponding to the operator $\hat A$, where

\[ { \sigma }_{ a}^{2} = \left < { A}^{ 2} \right > - { \left < A \right >}^{ 2} = \int { \psi }^{ *} { \hat A}^{ 2} \psi d \tau - { \left[ \int { \psi }^{ *} \hat A \psi d \tau \right ] }^{ 2} \]

Then the Uncertainty principle can be rigorously expressed as

\[ { \sigma }_{ a} { \sigma }_{ b} \geq \frac{ 1}{ 2} \left | \int { \psi }^{ *} \left[ \hat A, \hat B\right ] \psi d \tau \right | \]

In this case, the Uncertainty Principle for momentum and position can be written as

\[ { \sigma }_{ p} { \sigma }_{ x} \geq \frac{ 1}{ 2} \left | \int { \psi }^{ *} \left ( -i \hbar \right ) \psi d \tau \right | = \frac{ \hbar }{ 2} \]

  -- ThanhTruong - 12 Jul 2007

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