User:Brutulf/QM

Quantum Mechanics describes systems on a microscopic level where classical mechanics no longer apply. One of the main features of quantum mechanics is the fact that the energy levels of physical systems are quantisized, meaning that when energy enters or leaves a system (a molecule, for example) this happens only in discrete amounts, a 'quantum' of enery.

For example, the energy of any given photon is $E = h \cdot v$ where $h$ is Planck's Constant = $6,626 \cdot 10^-{34}{} \rm{Js}$, and $v$ is the frequency of the lightwave. Plancks constant is an experimentally determined quantity.

For isolated, microscopic systems the relative difference between energy levels may be significant, while macroscopic (large-scale) systems have miniscule differences between neighbouring energy levels. For macroscopic systems, the quantization of energy will therefore not by obvious, and the laws of classical mechanics apply very well. It can be said that quantum mechanics reduces to classical mechanics at the macroscopic level.

In contrast to classical mechanics, quantum mechanics do not lead to expressions for the exact position and velocity of the particles being considered. A particles behaviour is instead described by a mathematical function that gives a probability distribution for the values of postion, energy and other properties that the particle/system posesses.

At a quantum mechanic level, the behaviour of particles is stochastic.

As a given particle, for example an electron, can not be localized exactly from theory (it can be investigated experimentally, but it will not be possible to predict exactly what the result will be in each case), it is common to consider the particle as a delocalized phenomenon, or a wave in space. It is common to say that particles on a quantum mechanic level, 'have wave nature' or 'behave like waves'.

Because of this, the mathematical function that describes a particle is often called the wave function for the particle.

It may be neccesary to consider and treat a phenomenon both as both a particle and a wave. For example, light has properties that both implies that it consists of waves (diffraction, for example) and properties that implies that light it of particles (as in the photoelectric effect).

Key points:

• In the macroscopic world, objects can be treated as particles or clusters thereof with clearly defined and measurable locations and velocities.
• At a sufficiently microscopic level, particles must be considered quantum-mechanically as 'waves'.
• In general, a quantum mechanical particle has theoretically unclear position and velocities which can be considered only by way of a probability distribution
• The probability ditribution arises from a mathematical function called the wave function that describes the particle, and from which probabilities for the desired properties can be computed.
• Objects can have both particle and wave properties to varying extent

The wavelength of objects - De Broglie's wavelength

By combining Planck's equation $E = h \cdot v$ and Einstein's famous equation $E = mc^2$ the french scientist Louis de Broglie constructed the following expression for the wavelength of a particle: $\lambda = \frac{h}{mu} = \frac{h}{p}$ where h, again, is Planck's constant, m is the mass of the particle and u is the speed of the particle. On the rightmost side, the expression is written using p for the moment of the particle, where p = mu.

Example: An electron with a speed of $10^6 m/s$ and mass $9,11 \cdot 10^-31 kg$ has the wavelength $\lambda = \frac{h}{mu} = \frac{6,626 \cdot 10^-34 \rm{Js}}{9,11 \cdot 10^-31 \rm{kg} \cdot 10^6 \rm{m/s}} = 7,27 \cdot 10^-10 \rm{m}$

The result is of the same order of magnitude as the size of a small molecule, and the wave nature of the electron will therefore be of significance when considering such systems. A macroscopic object with mass 5 g travelling at a speed of 1 m/s will by contrast have a wavelength of about $10^-31$ m . Such a small wavelength for such a large object can not be measured experimentally, and the conclusion is that the macroscopic object has a negligible amount of 'wave nature' - it is fully adequate to describe it using classical mechanics.

In general: Quantum mechanics is used when the De Broglie-wavelength of a particle under consideration is abput equal to or greater than the characteristic size of the system.

The term characteristic size does not have an exact definition, but is related to the object of interest we are considering. For example, for the electron in the example, the de Broglie wavelenth was compared to the characteristic size of a small molecule with which the electron might be interacting. For the macroscopic particle, the characteristic wavelenth considered was the size of the particle itself (which was not declared, but which about reasonable assumptions can be made considering the mass of the particle).

Heisenberg's Uncertainty Relation

One of the most important results of quantum mechanics is Heisenbergs' uncertainty principle: To every quantum-mechanical quantity, there corresponds a complementary quantity. Two complementary quantities can never be determined completely precisely and simultanenously.

The best known example of this principle, is the uncertainty relation between the two quantities (a quantity being anything that can be measured) position and momentum:

$\Delta x \Delta x \ge \frac{h}{4 \pi} = \frac{\hbar}{2}$ where $\hbar$ is Planck's reduced constant $\hbar = \frac{h}{2\pi}$. $\delta x$ represtents the uncertainty in the position of the particle along the x axsis, while $\delta p$ represents the uncertainty in the moment of intertia of the particle along the same axis.

The uncertainty principle has a more general definition, on the same general form as the expression over, which requires some additional mathematics to be considered. Another specific example is the following relation, similar to the one above:

$\Delta E \Delta t \ge \frac{\hbar}{2}$ Here, E is the total energy of the system and t is time.

The Schrödinger Equation

As previously stated, when a particle is considered as a wave, the particle's behaviour can be described by a mathematical function. A wave function $\psi = \psi (x,y,z,t)$ is a function of position and time that describes such a particle.

The Schrödinger equation is a a differential equation with wave functions as solutions. If we constrain ourselves to x and t as variables, working in one space dimension only, the time-dependent Schrödinger equation can be written as:

$i\hbar\frac{{\partial \psi }}{{\partial t}} = - \frac{{{\hbar}^2 }}{{2m}}\frac{{\partial ^2 \psi }}{{\partial x^2 }} + V(x)\psi$

where V(x) is the potential enery of the particle at the point x (the potential is assumed to not change with time). i is the imaginary unit, $\sqrt { - 1}$, so the Schrödinger equation is a complex equation.

Justification:

If we c onsider a wave (an electromagnetic wave, for example) with amplitued $\psi$ that moves along the x-axis with the speed c, the wave can be described with the following wave equation:

$\frac{{\partial ^2 \psi }}{{\partial x^2 }} = \frac{1}{{c^2 }}\frac{{\partial ^2 \psi }}{{\partial t^2 }}$

$\psi (x,t)$ is the amplitude of the wave in the point x at time t.

A general solution of this equation can be written in complex form as:

$\psi (x,t) = Ce^{i2\pi (x/\lambda - vt)} = Ce^{i2\pi x/\lambda } e^{ - i2\pi vt}$ where C is a constant. If we replace $\lambda$, the wavelength, with $\frac{h}{p}$, from the de Broglie relation, and v with $\frac{E}{h}$ we get

$\psi (x,t) = Ce^{i2\pi xp/h} e^{ - i2\pi E/h} (*)$ Derivation with respect to t gives

$\frac{\partial }{{\partial t}}\psi (x,t) = - \frac{{i2\pi E}}{h}Ce^{i2\pi xp/h} e^{ - i2\pi E/h} = - \frac{{i2\pi E}}{h}\psi (x,t)$

we can now write $- \frac{h}{{2\pi i}}\frac{{\partial \psi }}{{\partial t}} = E\psi (**)$

Derivating (*) with respect to x we get:

$\frac{\partial }{{\partial x}}\psi (x,t) = \frac{{i2\pi p}}{h}Ce^{i2\pi xp/h} e^{ - i2\pi E/h} = \frac{{i2\pi p}}{h}\psi (x,t) (***)$

which gives

$\frac{h}{{2\pi i}}\frac{{\partial \psi }}{{\partial x}} = p\psi$

The total energy of the system equals the sum of kinetic and potential energy

$E = E_k + E_p = \frac{{p^2 }}{{2m}} + E_p (x)$

$(E_k = \frac{1}{2}mu^2 ,\,p = mu\, \to E_k = \frac{{p^2 }}{{2m}})$

Vi insert the expression for the enery into (**):

$- \frac{h}{{2\pi i}}\frac{{\partial \psi }}{{\partial t}} = \left[ {\frac{{p^2 }}{{2m}} + E_p (x)} \right]\psi = \frac{{p^2 }}{{2m}}\psi + E_p \psi (****)$

We now define the operator \[

$\hat H = \frac{{p^2 }}{{2m}} + E_p (x)$

From (***) we can extract the momentum operator:

$\hat p = \frac{h}{{2\pi i}}\frac{{\partial \psi }}{{\partial x}}$

Substituting $\hat H$ into (****), we have:

$\frac{{\partial ^2 \psi }}{{\partial x^2 }} = \frac{{i^2 4\pi ^2 p^2 }}{{h^2 }}\psi$

$\frac{{\partial \psi }}{{\partial t}} = - \frac{{i2\pi E}}{h}\psi$

$\begin{array}{l} \hat H\psi = \left[ { - \frac{{h^2 }}{{8\pi ^2 m}}\frac{{\partial ^2 }}{{\partial x^2 }} + E_p (x)} \right]\psi = - \frac{{h^2 }}{{8\pi ^2 m}}\frac{{i^2 4\pi ^2 p^2 }}{{h^2 }} + (E - \frac{{p^2 }}{{2m}})\left[ { - \frac{{h^2 }}{{8\pi ^2 m}}\frac{{i^2 4\pi ^2 p^2 }}{{h^2 }} + (E - \frac{{p^2 }}{{2m}})} \right]\psi \\ \,\,\,\,\,\,\,\,\, = \frac{{p^2 }}{{2m}}\psi + E\psi - \frac{{p^2 }}{{2m}}\psi = E\psi = - \frac{h}{{2\pi i}}\frac{{\partial \psi }}{{\partial t}} \\ \end{array}$

This is the one-dimensional, time-dependent Schrödinger equation.

In three dimensions, we replace $\frac{\partial }{{\partial x}}$ with the Laplace operator

$\nabla ^2 = \frac{{\partial ^2 }}{{\partial x^2 }} + \frac{{\partial ^2 }}{{\partial y^2 }} + \frac{{\partial ^2 }}{{\partial z^2 }}$ And define the Hamilton operator $\hat H$ as

$\hat H = - \frac{\hbar}{{2m}}\nabla ^2 + E_p$

We then get the time-dependent Schrödinger equation in three dimensions:

$\hat H\psi = - \frac{\hbar}{i}\frac{{\partial \psi }}{{\partial t}} = i\hbar\frac{{\partial \psi }}{{\partial t}}$

In a less compact form, the equation can be written as

$- \frac{{h^2 }}{{2m}}\nabla ^2 \psi + E_p \psi = ih\frac{{\partial \psi }}{{\partial t}}$

The set of solutions $\psi = \psi (x,y,z,t)$ of this equation describes possible states for a particle.

The time-dependent Schrödinger equation is a seprable differential equaion in space and time, and a general equation can be written as a product of two functions, one function in space and another in time:

$\Psi (x,y,z,t) = \psi (x,y,z,)\phi (t) = \psi (x,y,z,)e^{ - iEt/{\hbar}}$

Note: Even though the wave function varies with time, the square of the absolute value of the wave function does not:

$\left| {\Psi (\overrightarrow r ,t)} \right|^2 = \Psi ^* \Psi = \psi ^* e^{ - iEt/h} \psi e^{iEt/h} = \psi ^* \psi = \left| {\psi (\overrightarrow r )} \right|^2$

Separation of the Schrödinger equation

The (time-dependent) Schrödinger equation is a partial differential equation, which can be separated into two ordinary differential equations, one in time and one in space. The time-independent Schrödinger equation which is the ordinary differential equation which deals with position only:

$- \frac{{h^2 }}{{2m}}\nabla ^2 \psi + E_p \psi = E\psi$

Or more succintly

$\hat H\psi = E\psi \,\,\,\,\,\,\,\psi = \psi (x,y,z)$

The time-independent Schrödinger equation gives an infinite number of solutions with distinct values of E. The solutions are eigenfunctions of the Hamilton operator $\hat H$, and E is the corresponding eigenvalue. The solutions of the differential equation depend on the value of m and the potential V(x) and can be difficult or indeed impossible to solve for analytically.

In comparison, the solutions to the differential equation in time are much simpler. They are all of the form

$\phi (t) = e^{ - iEt/h}$

The general solution of the time-dependent Schrödinger equation is a linear combination of separable olutions. There is one wave equation for each allowed value of the energy E.

$\Psi _1 (\overrightarrow r ,t) = \psi _1 e^{ - iE_1 t/h} ,\Psi _2 (\overrightarrow r ,t) = \psi _1 e^{ - iE_2 t/\hbar} ...$

Every linear combination of solutions is itself a solution. We can therefore write a general solution as

$\Psi (\overrightarrow r ,t) = \sum\limits_{n = 1}^\infty {c_n } \psi _n (\overrightarrow r )e^{ - iE_n t/\hbar}$

All solutions of the time-dependent Schrödinger equation can be written on this form.

Example: A general type of problem might be (constraining ourselves to one space dimension): Given a potential V(x) and the value $\Psi (x,0)$ of the wave function over all space at time 0, find $\Psi (x,t)$.

Strategy: First, solve the time-independent Schrödinger equation. This gives an infinite number of solutions each with distinct values for E. The initial value criteria can be fulfilled by selecting suitable coefficients $c_n$ for

$\Psi (x,0) = \sum\limits_{n = 1}^\infty {c_n } \psi _n (x)$

$\Psi (x,t)$ is the constructed by multiplyin with the characteristic time-dependent exponential factor for each term.

$\Psi (x,t) = \sum\limits_{n = 1}^\infty {c_n } \psi _n (x)e^{ - iE_n t/h} = \sum\limits_{n = 1}^\infty {c_n } \Psi _n (x,t)$

Solution of the Schrödinger equation by separation of variables

We want to solve the time-dependent Schrödinger equation in space (here, x) and time (t).

$\hat H\Psi = ih\frac{{\partial \Psi }}{{\partial t}}$

or, writing out in full:

$- \frac{{h^2 }}{{2m}}\frac{{\partial ^2 \Psi }}{{\partial x^2 }} + V(x)\Psi = ih\frac{{\partial \Psi }}{{\partial t}}$

We assume that $\Psi (x,t)$ can be written as the product $\Psi (x,t) = \psi (x) \cdot \phi (t)$

We then get

$- \frac{{h^2 }}{{2m}}\psi ^{''} (x)\phi (t) + V(x)\psi \phi = ih\psi \phi ^'$

Dividing on both sides by $\psi \phi$ gives

$- \frac{{h^2 }}{{2m}}\frac{{\psi ^{''} }}{\psi } + V(x) = ih\frac{{\phi ^' }}{\phi } = E$ Both sides of the equation equal each other, but they depend on separate, independent variables. Therefore both sides must be constant, and equal to a separation constant which here is given the name E.

The differential equation for the variable x becomes:

$- \frac{{h^2 }}{{2m}}\frac{{\partial ^2 \psi }}{{\partial x^2 }} + (V - E)\psi = 0$

The potential V(x) must be specified before this equation can be solved.

The equation in the variable t becomes

$ih\frac{{\partial \phi }}{{\partial t}} - E\phi = 0$

$\frac{{\partial \phi }}{\phi } = \frac{E}{{ih}}\partial t \to \frac{{\partial \phi }}{\phi } = - \frac{{iE}}{h}\partial t$

$\int {\frac{{\partial \phi }}{\phi }} = \int { - \frac{{iE}}{h}\partial t} \to \ln \phi = - \frac{{iE}}{h}t + C$

$\phi = e^{ - \frac{{iE}}{h}t + C} = e^{ - \frac{{iE}}{h}t} e^C = Ce^{ - \frac{{iE}}{h}t}$

Absorbing the constant C into the solution of the equation in x, we get

$\underline{\underline {\Psi (x,t) = \psi (x)\phi (t) = \psi (x)e^{ - \frac{{iE}}{h}t} }}$

Postulates and principles

Some important postulates and principles of quantum mechanics are as follows:

Postulate: Every observable can be represented by a linear operator

Example: Kinetic energy can be represented with the operator $\hat E_k = - \frac{{h^2 }}{{2m}}\nabla ^2$

An operator $\hat o$ is linear if $\hat o(a\psi + b\varphi ) = a\hat o\psi + b\hat o\varphi$

Theorem: If two operators commute, every eigenfunction of one of the operators will also be an eigenfunction of the other operator.

Two operators commute if $\hat o_1 \hat o_2 \psi = \hat o_2 \hat o_1 \psi$

The commutator of two operators is defined as:

$\left[ {\hat o_1 ,\hat o_2 } \right] = \hat o_1 \hat o_2 - \hat o_2 \hat o_1$

Two operators commute if and only if their commutator is zero.

Probability distribution, interpretation of the wave function:

The value of the expression

$\psi ^* \psi \,dx\,dy\,dz = \left| \psi \right|^2 d\tau$

is proportional to the probability of finding the particle described by $\psi$ in the volume element $d\tau$

Restrictions on wave functions:

• The wave function must not go towards infinity over any area
• The wave function must have only one value for every point in space
• The wave function must be continueous, and its second derivative must be defined everywhere.
• The wave function must be quadratically integrable over its area of definition
• If $\psi ^* \psi d\tau$ is to give the actual probability of finding the particle in $d\tau$, the wave function must be normalized. That is:
$\int {\int {\int {\psi ^* \psi d\tau \,} } } = 1$

Expectation values:

Postulate: The expectation value <F> for a physical observable can be calculated from the following expression

$< F > \, = \int {\psi ^* } \hat F\psi \,d\tau$

It is assumed that $\psi$ is normalized. If $\psi$ is an eigenfunction of $\hat F$, with eigenvalue f, the eienvalue f is equal to the expectation value <F>.

Hermitean operators:

Observable quantities must be real (in the mathematical sense, contain no imaginary component). Therefore

$ = ^*$

This means that the corresponding operator $\hat F$ must be hermitean. For the operator to be hermitean, the following must be true:

$\int {\psi _1^* } \left[ {\hat F\psi _2 } \right]d\tau = \int {\left[ {\hat F\psi _1 } \right]} ^* \psi _2 \,d\tau$

Properties of hermitean operators:

• The eigenvalues of an hermitean operator are always real.
• The eigenfunctions that belong to distinct eigenvalues of an hermitean operator are always orthogonal.
Example: The Hamilton operator is hermitean and corresponds to an observable energy. If $\psi _1$ and $\psi _2$ correspond to different energies, it follows that they are orthogonal.

All observable quantities are represented by hermitean operators

Orthogonality:

That two wavefunctions $\psi _1$ and $\psi _2$ are orthogonal means that

$\int {\psi _1^* } \psi _2 \,d\tau = 0$

Degeneration: When two eigenfunctions share a common eigenvalue, that eigenvalue is degenerate. Example:

$\hat F\psi _1 = f\psi _1 \,\,\,\,\,\,\,\,\,\,\hat F\psi _2 = f\psi _2 \,$

Here, the eigenvalue f is two-degenerate.

If two eigenfunctions share the same eigenvalue, any linear combination of the two is also an eigenfunction. Example:

$\hat H(c_1 \psi _1 + c_2 \psi _2 ) = E(c_1 \psi _1 + c_2 \psi _2 )$