3.1 De Broglies Hypothesis | unit 3 quantum mechanics

PHY

Unit - 3

Quantum mechanics

3.1 De Broglie's Hypothesis

According to de-Broglie hypothesis, the energy of the universe is in the form of matter and radiation. So, both matter and radiation should have similar properties.

But radiation shows dual nature sometimes as wave and sometimes as particle. So, matter should have wave which can be expressed as:
Matter wave length = h/mv
where m is mass of particles/ matter
v= velocity and h is plank’s constant

Derivation of de-Broglie’s wavelength:
According to Plank’s quantum theory, energy of photon is expressed as

According to Einstein’s mass energy relation
de-Broglie’s wavelength

Now from equation i and ii we get
de-Broglie’s wavelength

This equation shows that wavelength depends upon mass and velocity of light that is momentum of the particle.

If the particle of a mass m moves with velocity v then the matter wavelength can be expressed as which is known as de-Broglie wavelength.

Hence the de-Broglie wavelength ranges between 0 and infinity.

3.2 Concept of Phase velocity and Group Velocity

Phase and group velocity are two important and related concepts in wave mechanics. They arise in quantum mechanics in the time development of the state function for the continuous case that is wave packets.

Harmonic Waves and Phase Velocity

A one-dimensional harmonic wave as shown in figure 1 is described by the equation,

where A0 is the wave amplitude,

w is the circular frequency;

k is the wave number;

and is an initial, constant phase.

Sometimes the wave number is referred to as the spatial frequency or propagation constant.

Figure 1: Harmonic Wave

This is a monochromatic wave of one frequency. There are no strictly monochromatic waves in nature.

For example, the generating source of the wave may move slightly, introducing spurious frequencies.

In general, these waves propagate without warping. That is, the phase (x, t) is a constant:

vphase is the phase velocity for a wave.

For sending information, these waves are not useful because they are the same throughout time and space. Some must be modulated, such as frequency or amplitude, in order to convey information.
The resulting wave may be a perturbation that acts over a short distance, that is a wave packet. This wave packet can be considered to be a superposition of a number of harmonic waves, that is in other words a Fourier series or integral.

Group Velocity

In order to convey information, it is essential to have a thing more than a simple harmonic wave is required. However, the superposition of many such waves of varying frequencies can result in an "envelope" wave and a carrier wave within the envelope.

The envelope can transmit data.

A simple example is the superposition of two harmonic waves with frequencies that are very close (w1 ~w2) and of the same amplitude. The equations for the motion are,

The plot of such a wave is shown in Figure 2.

Figure 2: Group Velocity

The envelope (the green line) is given by u1 and travels at the group velocity. The carrier wave (the blue line) travels at the phase velocity and is given by u2. The wave packet moves at the group velocity. It is the envelope which carries information. Group velocity is given by,

Phase and group velocity are related through Rayleigh's formula,

If the derivative term is zero, group velocity equals phase velocity.

In this case, there is no dispersion.

Dispersion is when the distinct phase velocities of the components of the envelope cause the wave packet to "spread out" over time.

3.3 Heisenberg uncertainty principle

The observables have only discrete sets of experimental values. For example, the values of the energy of a bound system are always discrete, and angular momentum components have values that take the form mℏ, where m is either an integer or a half-integer, positive or negative.
The position of a particle or the linear momentum of a free particle can take continuous values in both quantum and classical theory. The mathematics of observables with a continuous spectrum of measured values is somewhat more complicated than for the discrete case but presents no problems of principle.
An observable with a continuous spectrum of measured values has an infinite number of state functions. The state function Ψ of the system is regarded as a combination of the state functions of the observable, but the sum has to be replaced by an integral.
The uncertainty principle is significant only on the atomic scale because of the small value of h in everyday units.
If the position of a macroscopic object with a mass of, say, one gram is measured with a precision of 10−6 metre, the uncertainty principle states that its velocity cannot be measured to better than about 10−25 metre per second.
However, if an electron is located in an atom about 10−10 metre across, the principle gives a minimum uncertainty in the velocity of about 106 metre per second.

3.4 Wave function and its physical significance

Wave function is a variable quantity that mathematically describes the wave characteristics of a particle.
The value of the wave function of a particle at a given point of space and time is related to the likelihood of the particle being there at the time.
By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave.
The square of the wave function, Ψ2, however, does have physical significance: the probability of finding the particle described by a specific wave function Ψ at a given point and time is proportional to the value of Ψ2.

3.5 Schrodinger Equation:

Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation cannot be derived using elementary methods and is generally given as a postulate of quantum mechanics.

The single-particle three-dimensional time-dependent Schrödinger equation is

$\begin{displaymath} i \hbar \frac{\partial \psi({\bf r},t)}{\partial t} = - \f... ...r^2}{2m} \nabla^2 \psi({\bf r},t) + V({\bf r}) \psi({\bf r},t) \end{displaymath}$

(1)

where $V$ is assumed to be a real function and represents the potential energy of the system.

Of course, the time-dependent equation can be used to derive the time-independent equation. If we write the wavefunction as a product of spatial and temporal terms, $\psi({\bf r}, t) = \psi({\bf r}) f(t)$ , then equation (1) becomes

$\begin{displaymath} \psi({\bf r}) i \hbar \frac{df(t)}{dt} = f(t) \left[ - \frac{\hbar^2}{2m} \nabla^2 + V({\bf r}) \right] \psi({\bf r}) \end{displaymath}$

(2)

$\begin{displaymath} \frac{i \hbar}{f(t)} \frac{df}{dt} = \frac{1}{\psi({\bf r})}... ...frac{\hbar^2}{2m} \nabla^2 + V({\bf r}) \right] \psi({\bf r}) \end{displaymath}$

(3)

Since the left-hand side is a function of $t$ only and the right-hand side is a function of ${\bf r}$ only, the two sides must equal a constant.

If we tentatively designate this constant $E$ then we extract two ordinary differential equations, namely

$\begin{displaymath} \frac{1}{f(t)} \frac{df(t)}{dt} = - \frac{i E}{\hbar} \end{displaymath}$

(4)

and

$\begin{displaymath} -\frac{\hbar^2}{2m} \nabla^2\psi({\bf r}) + V({\bf r}) \psi({\bf r}) = E \psi({\bf r}) \end{displaymath}$

(5)

The latter equation is once again the time-independent Schrödinger equation.

Theformer equation is easily solved to yield

$\begin{displaymath} f(t) = e^{-iEt / \hbar} \end{displaymath}$

(6)

The Hamiltonian in equation (5) is a Hermitian operator, and the eigenvalues of a Hermitian operator must be real, so $E$ is real. This means that the solutions $f(t)$ are purely oscillatory, since $f(t)$ never changes in magnitude

(recall Euler's formula $e^{\pm i \theta} = cos \theta \pm i \hspace{5pt} sin \theta$ ).

Thus if

$\begin{displaymath} \psi({\bf r}, t) = \psi({\bf r}) e^{-iEt / \hbar} \end{displaymath}$

(7)

then the total wave function $\psi({\bf r}, t)$ differs from $\psi({\bf r})$ only by a phase factor of constant magnitude.

There are some interesting consequences of this. First of all, the quantity $\vert \psi({\bf r}, t) \vert^2$ is time independent, as we can easily show:

$\begin{displaymath} \vert \psi({\bf r}, t) \vert^2 = \psi^{*}({\bf r}, t) \psi({... ...{-iEt / \hbar} \psi({\bf r}) = \psi^{*}({\bf r}) \psi({\bf r}) \end{displaymath}$

(8)

Secondly, the expectation value for any time-independent operator is also time-independent, if $\psi({\bf r}, t)$ satisfies equation (7). By the same reasoning applied above,

$\begin{displaymath} <A> = \int \psi^{*}({\bf r}, t) \hat{A} \psi({\bf r}, t) = \int \psi^{*}({\bf r}) \hat{A} \psi({\bf r}) \end{displaymath}$

(9)

For these reasons, wave functions of the form (7) are called stationary states. The state $\psi({\bf r}, t)$ is stationary,'' but the particle it describes is not!

Equation (7) represents a particular solution to equation (1).

The general solution to equation (1) will be a linear combination of these particular solutions, that is

$\begin{displaymath} \psi({\bf r}, t) = \sum_i c_i e^{-iE_it / \hbar} \psi_i({\bf r}) \end{displaymath}$

The Time-Independent Schrödinger Equation

The one-dimensional classical wave equation,

$\begin{displaymath} \frac{\partial^2u}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2u}{\partial t^2} \end{displaymath}$

(1)

By introducing the separation of variables

$\begin{displaymath} u(x,t) = \psi(x)f(t) \end{displaymath}$

(2)

we obtain

$\begin{displaymath} f(t) \frac{d^2 \psi(x)}{dx^2} = \frac{1}{v^2} \psi(x) \frac{d^2f(t)}{dt^2} \end{displaymath}$

(3)

If we introduce one of the standard wave equation solutions for $f(t)$ such as $e^{i\omega t}$ we obtain

$\begin{displaymath} \frac{d^2\psi(x)}{dx^2} = \frac{-\omega^2}{v^2} \psi(x) \end{displaymath}$

(4)

Now we have an ordinary differential equation describing the spatial amplitude of the matter wave as a function of position. The energy of a particle is the sum of kinetic and potential parts

$\begin{displaymath} E = \frac{p^2}{2m} + V(x) \end{displaymath}$

(5)

which can be solved for the momentum, $p$ , to obtain

$\begin{displaymath} p = \{ 2m [ E - V(x) ] \} ^{1/2} \end{displaymath}$

(6)

Now we can use the de Broglie formula to get an expression for the wavelength

$\begin{displaymath} \lambda = \frac{h}{p} = \frac{h}{\{ 2m [ E - V(x) ] \}^{1/2}} \end{displaymath}$

(7)

The term $\omega^2/v^2$ in equation (4) can be rewritten in terms of $\lambda$ if we recall that $\omega = 2 \pi \nu$ and $\nu \lambda = v$ .

$\begin{displaymath} \frac{\omega^2}{v^2} = \frac{4 \pi^2 \nu^2}{v^2} = \frac{4\pi^2}{\lambda^2} = \frac{2m[E - V(x)]}{\hbar^2} \end{displaymath}$

(8)

When this result is substituted into equation (4) we obtain the famous time-independent Schrödinger equation

$\begin{displaymath} \frac{d^2\psi(x)}{dx^2} + \frac{2m}{\hbar^2} [ E - V(x)]\psi(x) = 0 \end{displaymath}$

(9)

which is almost always written in the form

$\begin{displaymath} -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \end{displaymath}$

(10)

This single-particle one-dimensional equation can easily be extended to the case of three dimensions, where it becomes

$\begin{displaymath} -\frac{\hbar^2}{2m} \nabla^2\psi({\bf r}) + V({\bf r}) \psi({\bf r}) = E \psi({\bf r}) \end{displaymath}$

(11)

A two-body problem can also be treated by this equation if the mass $m$ is replaced with a reduced mass $\mu$ .

3.6 Application of Schrodinger equation:

Consider one dimensional closed box of width L. A particle of mass ‘m’ is moving in a one-dimensional region along X-axis specified by the limits x=0 and x=L as shown in fig. The potential energy of particle inside the box is zero and infinity elsewhere.

Potential energy V(x) is of the form

V(x) = {o; if o<x<L

∞: elsewhere

The one-dimensional time independent Schrodinger wave equation is given by

d2ψ/dx2+ 2m/Ћ2[E-V] ψ=0 (1)

Here the partial derivatives have been changed because equation now contains only one variable that is x-Co-ordinate. Inside the box V(x) =0

Therefore, the Schrodinger equation in this region becomes

d2/ψ/dx2+ 2m/Ћ2Eψ=0

Or d2ψ/dx2+ K2ψ=0 (2)

Where k= 2mE/Ћ2 (3)

k is called the Propagation constant of the wave associated with particle and it has dimensions reciprocal of length.

The general solution of eq (2) is

Ψ=A sin Kx + B cos K x (4)

Where A and B are arbitrary conditions and these will be determined by the boundary conditions.

(ii) Boundary Conditions

The particle will always remain inside the box because of infinite potential barrier at the walls. So, the probability of finding the particle outside the box is zero that is .ψx=0 outside the box.

We know that the wave function must be continuous at the boundaries of potential well at x=0 and x=L, i.e.

Ψ(x)=0 at x=0 (5)

Ψ(x)=0 at x= L (6)

These equations are known as Boundary conditions.

(iii) Determination of Energy of Particle

Apply Boundary condition of eq.(5) to eq.(4)

0=A sin (X*0) +B cos (K*0)

0= 0+B*1

B=0 (7)

Therefore eq.(4) becomes

Ψ(x) = A sin Kx (8)

Applying the boundary condition of eq.(6) to eq.(8) ,we have

0=A sin KL

Sin KL=0

KL=nπ

K=nπ/L (9)

Where n= 1, 2, 3 – – –

A Cannot be zero in eq. (9) because then both A and B would be zero. This will give a zero wave function every where which means particle is not inside the box.

Wave functions. Substitute the value of K from eq. (9) in eq. (8) to get

Ψ(x)=A sin(nπ/Lx)

As the wave function depends on quantum number π so we write it ψn. Thus

Ψn=A sin (nπx/L)0<x<L

This is the wave function or eigen function of the particle in a box.

Ψn=0 outside the box

Energy value or Eigen value of particle in a box: Put this value of K from equation (9) in eq. (3)

nπ/L = 2m E/Ћ2

Squaring both sides

n2π2/L2=2mE/Ћ2

E=n2π2Ћ2/2mL2

Where n= 1, 2, 3… Is called the Quantum number

As E depends on n, we shall denote the energy of particle arEn. Thus

En= n2π2Ћ2/2mL2 (10)

This is the eigen value or energy value of the particle in a box.

3.7 TUNNELING EFFECT

In classical mechanics, to describe a system of material point at a certain moment of time, it is enough to set every point coordinates and momentum components.
Using the principle of quantum mechanics, itis impossible to determine simultaneously coordinates and momentum components of even single point according to the He-isenberg uncertainty principle.
In order to describe the system completely, an associated complex function is introduced in quantum mechanics.
The wavefunction , a function of time and all system particles position, is a solution of the wave Schrodinger equation.
In order to use the system wavefunction, one should determine rather than . Then, the probability for finding particles in an elementary volume dxdydz is given by .

Let’s calculate the transparency of the rectangular barrier [1, 2]. Suppose that electrons of potential energy

(1)

impinge on the rectangular potential barrier and the total energy E is less than U0 (Fig. 1).

Fig.3. Rectangular potential barrier and particle wave function .

The stationary Schrodinger equations can be written as follows

(2)

where , – wave vectors, – Planck's constant.

The solution to the wave equation at can be expressed as a sum of incident and reflected waves , while solution at – as a transmitted wave .

A general solution inside the potential barrier is written as . Constants a, b, c, d is determined from the wavefunction and continuity condition at and .

The barrier transmission coefficient can be naturally considered as a ratio of the transmitted electrons probability flux density to that one of the incident electrons.

In the case under consideration this ratio is just equal to the squared wavefunction module at because the incident wave amplitude is assumed to be 1 and wave vectors of both incident and transmitted waves coincide.

(3)

If , then both and can be approximated to and (3) will be written as

(4)

where .

Thus, analytical calculation of the rectangular barrier transmission coefficient is rather a simple task.

Tunneling effect examples

Alpha decay

Alpha decay represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. This transition can be characterized as:

Figure 4. Alpha Decay

As can be seen from the figure, alpha particle is emitted in alpha decay.Among the variety of channels in which a nucleus decays, alpha decay has been one of the most studied.

The alpha decay channel in heavy and super heavy nuclei which provide information on the fundamental properties of nuclei far from stability, such as their ground state energies and the structure of their nuclear levels.

Alpha decay is a quantum tunnelling process. In order to be emitted, the alpha particle must penetrate a potential barrier. This is similar to cluster decay, in which an atomic nucleus emits a small “cluster” of neutrons and protons (e.g. 12C).

Fig 5. Graph

A scanning tunnelling microscope (STM) is a non-optical microscope that works by scanning an electrical probe tip over the surface of a sample at a constant spacing.

 STM

The STM sample must conduct electricity for the process to work. The STM uses a tip that ends in a single atom, and a voltage is passed through the tip and the sample.

Electrons use a quantum mechanical effect to ‘tunnel’ from the tip to the sample and vice versa. The current that results depends upon the distance between the probe tip and the sample surface.

The tip is attached to a piezoelectric tube, and the voltage applied to the piezo rod is altered to maintain a constant distance of the tip from the surface. The changes in this voltage allow a three-dimensional picture of the material surface to be built up as the tip is scanned back and forth across the sample.

Figure6. STM tip

Tunnel diode

A Tunnel diode is a heavily doped p-n junction diode in which the electric current decreases as the voltage increases.

In tunnel diode, electric current is caused by “Tunnelling”. The tunnel diode is used as a fast switching device in computers. It is also used in high-frequency oscillators and amplifiers.

In tunnel diode, the p-type semiconductor act as an anode and the n-type semiconductor act as a cathode.

Figure 7. Tunnel doide

The anode is a positively charged electrode which attracts electrons whereas cathode is a negatively charged electrode which emits electrons.
In tunnel diode, n-type semiconductor emits or produces electrons so it is referred to as the cathode.
On the other hand, p-type semiconductor attracts electrons emitted from the n-type semiconductor so p-type semiconductor is referred to as the anode.

Construction:

If a semiconductor diode is heavily doped with impurities, it will exhibit negative resistance. Negative resistance means the current across the tunnel diode decreases when the voltage increases.
The operation of tunnel diode depends on the quantum mechanics principle known as “Tunnelling”.
In electronics, tunnelling means a direct flow of electrons across the small depletion region from n-side conduction band to the p-side valence band.

Fig 8. PN junction

The germanium material is commonly used to make the tunnel diodes. They are also made from other types of materials such as gallium arsenide, gallium antimonide, and silicon.

3.8 Introduction to quantum computing

Computers are getting smaller and faster day by day because electronic components are getting smaller and smaller. However, this process may meet its physical limit.
Electricity is flow of electrons. Since size of transistors is shrinking to size of few atoms, transistors cannot be used as switch because electron may transfer themselves to the other side of blocked passage by the process called quantum tunnelling.
Quantum mechanics is a branch of physics that explores physical world at most fundamental level.
At this level particle behave differently from classical world taking more than one state at the same time and interacting with other particles that are very far away. Phenomena like superposition and entanglement take place.
Quantum computers can easily crack the encryption algorithms used today in very less time. They have great solving optimization problems.

Reference:

1. Engineering Physics-Avadhanulu, Kshirsagar, S. Chand Publications

2. A textbook of optics- N Subrahmanyam and BriLal , S. Chand Publications

3. Engineering Physics- Gaur, Gupta, Dhanpat Rai and Sons Publications

Sign Up

Index

Notes

Highlighted

Underlined

Browse by Topics

Notes

Highlighted

Underlined