Quantum Mechanics

The term "Quantum Mechanics" was first coined by the German physicist Max Born in 1924. It is a fundamental theory in physics that describes the behavior of particles at the smallest scales, such as atoms, electrons, photons, and other subatomic particles. It is a crucial part of modern physics, providing the foundation for understanding the physical properties of matter and energy at microscopic scales.

Dual Nature of Light - Certain phenomena, such as interference, diffraction, and polarization, can be explained by considering light as a wave. On the other hand, phenomena such as the photoelectric effect and the Compton effect can be explained by considering light as a particle. When we visualize light as a wave, we need to forget its particle aspect completely, and vice versa. This type of behavior, where light exhibits properties of both waves and particles, is known as the dual nature of light.

De Broglie Waves

In 1924, the French physicist Louis de Broglie proposed that particles of matter, such as electrons, also exhibit wave-like properties. This revolutionary idea extended the concept of wave-particle duality, which was initially applied to light, to all forms of matter. De Broglie hypothesized that every particle with momentum has an associated wavelength, known as the de Broglie wavelength. The de Broglie wavelength (𝜆) of a particle is given by the following equation:

$\lambda = \frac{h}{p}$

Where:

• $\lambda$ is the de Broglie wavelength
• $h$ is Planck's constant ($6.626 \times 10^{-34} Js$),
• $p$ is the momentum of the particle.

Expression for De Broglie Wavelength

The energy $E$ of a photon can be expressed in terms of its frequency $\nu$ as:

$E= h \nu \; \; \; \text{----- eq 1}$

Where $h$ is Planck's constant.

and the frequency $\nu$ of a photon is related to its wavelength 𝜆 and the speed of light c by the equation:

$\nu = \frac{c}{\lambda} \; \; \; \text{----- eq 2}$

Substituting eq 2 in eq 1, we get:

$E = h\left(\frac{c}{\lambda}\right) \; \; \; \text{----- eq 3}$

According to Einstein's theory of relativity, the energy $E$ of a particle with mass $m$ is given by:

$E=mc^2 \; \; \; \text{----- eq 4}$

On equating eq 3 and eq 4, we get:

$mc^2 = \frac{hc}{\lambda}$

$\therefore \; \; \lambda = \frac{h}{mc} = \frac{h}{p}$

where $p$ is momentum of particle

De-Broglie Wavelength in terms of Energy

We know Kinetic Energy $E$ of particle is,

$E = \frac{1}{2}m\nu^2$

On dividing and multiplying RHS of above equation with $m$ we get,

$E = \frac{m^2\nu^2}{2m}$

$m^2\nu^2 = 2mE$

$p^2 = 2mE$

$p = \sqrt{2mE} \; \; \; \text{----- eq A}$

We know,

$\lambda = \frac{h}{p} \; \; \; \text{----- eq B}$

Substituting eq A in eq B we get,

$\lambda = \frac{h}{\sqrt{2mE}}$

This is the expression for de-Broglie wavelength in terms of energy.

If velocity $v$ is given to an electron by accelerating it through a potential difference $V$, then the work done on the electron is $eV$. This work done is converted to kinetic energy of electron. Hence, we can write,

$\frac{1}{2}mv^2 = eV$

so the expression $\lambda = \frac{h}{\sqrt{2mE}}$ becomes,

$\lambda = \frac{h}{\sqrt{2meV}}$

This is of de Broglie wavelength of an accelerated electron.

Properties of Matter Waves

1. As lighter the particle, greater is its wavelength.
2. Smaller is the velocity of particle, greater is its wavelength.
3. If velocity = 0 then $\lambda$ is infinite i.e. matter waves are generated only with moving particles
4. Matter waves do not require a medium for propagation and can travel through a vacuum.
5. Velocity of matter waves depend on velocity of matter particles that means it is not constant.

Phase and Group Velocity Concept

Phase Velocity

Phase velocity is the rate at which the phase of a wave propagates through space. It is particularly relevant for monochromatic waves (single frequency waves) and represents the speed at which the crests and troughs of the wave move. The phase velocity is significant in understanding how individual wavefronts progress in various media.

Expression for Phase Velocity

We can write the deBroglie wave travelling along the +x direction as,

$$y = a\: sin(\omega t - kx) \quad \text{(i)}$$

where $a$ is the amplitude, $(w = 2 \pi v)$ is the angular frequency and $(k= \frac{2 \pi}{\lambda})$ is the propagation constant of the wave. By the definition, the ratio of angular frequency w to the propagation constant $k$ is the phase (or wave) velocity. If we represent the phase velocity by $V_p$ then,

$$V_p = \frac{\omega}{k}$$

The phase of the wave motion is given by $(\omega t - kx)$. It means the particle of constant phase travels such that $(\omega t - kx)$ = constant.

Taking the derivative with respect to time, we get:

$$\frac{d(\omega t - kx)}{dt} = 0$$ $$\omega - k\frac{dx}{dt} = 0$$

Thus:

$$\frac{dx}{dt} = V_p = \frac{\omega}{k} \quad \text{(ii)}$$

where $V_p = \frac{dx}{dt}$ is the phase (or wave) velocity. Thus, the wave velocity is the velocity of planes of constant phase which advance through the medium. We can write the phase velocity $V_p = v \lambda$ and for an electromagnetic wave $E = h v$, or $v = \frac{E}{h}$

According to de Broglie:

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$

Therefore:

$$V_p = v \lambda = \frac{E}{h} \times \frac{h}{mv} = \frac{mc^2}{mv} = \frac{c^2}{v}$$

Thus:

$$V_p = \frac{c^2}{v} \quad \text{(iii)}$$

Since $c >> v$, Eq. (iii) implies that the phase velocity of the de Broglie wave associated with a particle moving with velocity $v$ is greater than $c$, the speed of light.

Group Velocity

Group velocity is the rate at which the envelope of a wave packet or pulse travels through space. It is particularly relevant for wave packets composed of multiple frequency components and represents the speed at which information or energy is conveyed by the wave packet.

In a dispersive medium, where the phase velocity varies with frequency, the group velocity can differ from the phase velocity. The group velocity is crucial in understanding the propagation of signals and energy in such media.

Expression for Group Velocity

As we have seen, the phase velocity of a wave associated with a particle comes out to be greater than the velocity of light. This difficulty can be overcome by assuming that each moving particle is associated with a group of waves or a wave packet rather than a single wave. In this context, de Broglie waves are represented by a wave packet, and hence we have group velocity associated with them. In order to understand the concept of group velocity, we consider the combination of two waves, the resultant of which is shown in the figure below.

The two waves are represented by the following relations:

$$y_1 = a \sin(\omega_1t - k_1x ) \quad \text{(i)}$$ $$y_2 = a \sin(\omega_2t - k_2x ) \quad \text{(ii)}$$

Their superposition gives:

$$y = y_1 + y_2$$

and

$$y_1 + y_2 = a[\sin(\omega_1t - k_1x ) + \sin(\omega_2t - k_2x )]$$

or

$$y = 2a \sin \left[\frac{(\omega_1 + \omega_2)t}{2} - \frac{(k_1 + k_2)x}{2}\right] \cdot$$ $$\cos\left[\frac{(\omega_1 + \omega_2)t}{2} - \frac{(k_1 + k_2)x}{2}\right]$$

Therefore,

$$y = 2a \cos \left[\frac{(\omega_1 - \omega_2)t}{2} - \frac{(k_1 - k_2)x}{2}\right] \cdot$$ $$\sin(\omega t - kx) \quad \text{(iii)}$$

where

$$\omega = \frac{\omega_1 + \omega_2}{2}, k = \frac{k_1 + k_2}{2}$$

Eq. (iii) can be rewritten as:

$$y = 2a cos \left[\frac{(\Delta \omega)t}{2} - \frac{(\Delta k)x}{2}\right] \sin(\omega t - kx)$$

where $\Delta \omega = \omega_1 - \omega_2$ and $\Delta k = k_1 - k_2$.

The resultant wave Eq. (iv) has two parts:

1. A wave of frequency $\omega$, propagation constant k, and velocity u, given by

   $$V_p = \frac{\omega}{k} = \frac{2 \pi v}{\frac{2 \pi}{\lambda}} = v \lambda$$

   which is the phase velocity or wave velocity.

2. Another wave of frequency $\frac{\Delta \omega}{2}$, propagation constant $\frac{\Delta k}{2}$, and the velocity $V_g$, given by

   $$V_g = \frac{\Delta \omega}{\Delta k}$$

   This velocity is the velocity of the envelope of the group of waves, i.e., it is the velocity of the wave packet (shown by dotted lines) and is known as the group velocity. For the waves having a small difference in their frequencies and wave numbers, we can write

   $$V_g = \frac{\Delta \omega }{\Delta k} = \frac{\partial \omega }{\partial k} = \frac{\partial (2 \pi v)}{\partial (\frac{2 \pi}{\lambda})} = \frac{\partial v}{\partial \frac{1}{\lambda}}$$

   and

   $$\frac{\partial v}{\partial \frac{1}{\lambda}} = -\lambda^2 \frac{\partial v}{\partial \lambda}$$ $$\therefore \; V_g = - \frac{\lambda^2 \partial \omega}{2 \pi \partial \lambda}$$

   This is the expression for the group velocity.

Difference Betwwen Phase and Group Velocity

Relation between Phase and Group Velocity

If $V_p$ be the phase (wave) velocity, then the group velocity can be written as,

$$V_g = \frac{d \omega}{dk} = \frac{d}{dk}(V_pk)$$

Since,

$$V_p = \frac{\omega}{k}$$

or

$$V_g = V_p + k\frac{dV_p}{dk}$$

But $k = \frac{2 \pi}{\lambda}$ $\implies$ $dk = -\frac{2 \pi}{\lambda^2}d \lambda$

So,

$\frac{k}{dk} = -\frac{\lambda}{d \lambda}$

Therefore, the group velocity is given by

$$V_g = V_p + (-\frac{\lambda}{d \lambda})dV_p$$

or

$$V_g = V_p - \lambda \frac{dV_p}{d \lambda}$$

The relation shows that the group velocity $V_g$ is less than the phase velocity $V_p$ in a dispersive medium where $V_p$ is a function of $k$ or $\lambda$. In a non-dispersive medium, the velocity $V_p$ is independent of $k$ or $\lambda$, meaning $\frac{dV_p}{d \lambda} = 0$, thus $V_g = V_p$. This is true for electromagnetic waves in a vacuum and elastic waves in a homogeneous medium.

Uncertainty Principle

The Uncertainty Principle, formulated by Werner Heisenberg in 1927, is a fundamental concept in quantum mechanics that states certain pairs of physical properties, like position and momentum, cannot be precisely determined simultaneously.

1. Principle Statement:

   The Uncertainty Principle asserts that the more precisely you know the position of a particle, the less precisely you can know its momentum, and vice versa. Mathematically, it's often expressed as:

   $$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$

   where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and ℏ (reduced Planck's constant) is a fundamental constant.

2. Physical Interpretation:

   This principle arises due to the wave-like properties of particles in quantum mechanics. A particle's position is associated with a wave function that spreads out over space, while momentum is related to the wave's wavelength and therefore its spatial extent. Trying to precisely locate a particle requires a wave packet that is very localized in space, which results in a broad range of possible wavelengths (and thus momenta).

Heisenberg’s γ-ray Microscope

Heisenberg’s gamma-ray microscope is a theoretical concept proposed by physicist Werner Heisenberg in 1927. The idea behind the microscope is to use gamma-ray photons to visualize objects with extremely high resolution, potentially even at the atomic scale.

The concept relies on the uncertainty principle, which Heisenberg himself formulated. According to this principle, there is a limit to the precision with which certain pairs of physical properties, such as the position and momentum of a particle, can be simultaneously known. In the case of the gamma-ray microscope, the uncertainty principle imposes a limitation on the accuracy with which one can determine the position of the particle being observed.

Heisenberg proposed that by using gamma-ray photons with extremely short wavelengths, it would be possible to confine the location of the object being observed within a very small volume. This confinement would be achieved by scattering the gamma-ray photons off the object, which would cause the position of the object to become uncertain due to the momentum transfer from the photons. By measuring the scattered gamma rays, one could gather information about the object’s position with high precision.

Expression for Heisenberg Uncertainty Principle

Heisenberg’s uncertainty principle can be proved on the basis of de Broglie’s wave concept that a material particle in motion is equivalent to a group of waves or wave packet, the group velocity $G$ being equal to the particle velocity $v$. Consider a simple case of a wave packet which is formed by the superposition of two simple harmonic plane waves of equal amplitudes $a$ and having nearly equal frequencies $\omega_1$ and $\omega_2$. The two waves can be represented by the equations:

$$y_1 = a sin(\omega_1t - k_1x)$$ $$y_2 = a sin(\omega_2t - k_2x)$$

where $k_1$ and $k_2$ are their propagation constants and $\frac{\omega_1}{k_1}$ and $\frac{\omega_2}{k_2}$ are their respective phase velocities. The resultant wave due to superposition of these waves is given by:

$$y = y_1 + y_2$$ $$y = 2a \sin (\omega t - kx) \cos \left[ \frac{\Delta \omega}{2} t - \frac{\Delta k}{2} x \right] \quad \text{(i)}$$

where $\omega = (\omega_1 + \omega_2)/2$, $k = (k_1 + k_2)/2$, $\Delta \omega = \omega_1 - \omega_2$ and $\Delta k = k_1 - k_2$.

The resultant wave is shown in Fig. The envelope (loop) of this wave travels with the group velocity $G$, given by

$$G = \frac{\Delta \omega}{\Delta k} = \frac{\omega_1 - \omega_2}{k_1 - k_2}$$

Since the group velocity of deBroglie wave group associated with the moving particle is equal to the particle velocity, the loop so formed is equivalent to the position of the particle. Then the particle may be anywhere within the loop. Now the condition of the formation of node from Eq. (i) is given by

$$\cos \left[ \frac{\Delta \omega}{2} t - \frac{\Delta k}{2} x \right] = 0$$

or

$$\frac{\Delta \omega}{2} t - \frac{\Delta k}{2} x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots, \frac{(2n+1)\pi}{2} \quad \text{(ii)}$$

where $n = 0, 1, 2, \ldots$

If $x_1$ and $x_2$ be the values of positions of two consecutive nodes, then from above equation by putting $n$ and $(n+1)$, we get

$$\frac{\Delta \omega}{2} t - \frac{\Delta k}{2} x_1 = \frac{(2n+1)\pi}{2} \quad \text{(iii)}$$

and

$$\frac{\Delta \omega}{2} t - \frac{\Delta k}{2} x_2 = \frac{(2n+3)\pi}{2} \quad \text{(iv)}$$

From Eqs. (iii) and (iv), we have

$$\frac{\Delta k}{2}(x_1 - x_2) = \pi$$

or

$$\frac{\Delta k}{2} \Delta x = \pi \quad \text{(v)}$$

or

$$\Delta x = \frac{2\pi}{\Delta k}$$

but

$$k = \frac{2\pi}{\lambda} = \frac{2\pi}{h/p} = \frac{2\pi p}{h}$$ $$\Delta k = \frac{2\pi}{h} \Delta p$$

where $\Delta p$ is the error (uncertainty) in the measurement of momentum $p$. Therefore, from Eq. (v)

$$\Delta x = \frac{2\pi h}{2\pi \Delta p} = \frac{h}{\Delta p}$$

or

$$\Delta p \Delta x = h$$

However, more accurate measurements show that the product of uncertainties in momentum ($\Delta p$) and the position ($\Delta x$) cannot be less than $h/2\pi$. Therefore

$$\Delta p \Delta x \geq \hbar$$

This is the Heisenberg’s uncertainty principle.

Heisenberg Uncertainty Principle Application

The Heisenberg Uncertainty Principle is used in several fields of physics, technology and even philosophy nowadays. Here are some notable applications:

1. Quantum Mechanics: The Uncertainty Principle is a fundamental concept in quantum mechanics. It describes the probabilistic nature of quantum systems and highlights the limitations of classical physics in predicting the behavior of particles.

2. Atomic and Molecular Physics: In atomic and molecular physics, the Uncertainty Principle helps understand the behavior of electrons within atoms and molecules. It provides insights into electron orbitals, energy levels, and chemical bonding, explaining the stability and structure of atoms and molecules.

3. Quantum Computing: Quantum computers utilize the principles of quantum mechanics, including the Uncertainty Principle, to perform computations using quantum bits (qubits).

4. Electron Microscopy: In electron microscopy, the Uncertainty Principle sets limits on the spatial resolution of images obtained using electron beams. It determines the minimum size of features that can be resolved in a sample.

5. Particle Physics: In particle physics, the Uncertainty Principle influences the study of subatomic particles and their interactions. It provides insights into phenomena such as particle decay, scattering processes, and the uncertainty in the measurement of particle properties.