Unit IV Quantum mechanics| PHY 110 Engineering Physics |B.tech

   

 Quantum mechanics 

1. Need for Quantum Mechanics

Quantum mechanics arose in the early 20th century to address the limitations of classical physics when explaining phenomena at atomic and subatomic levels. Classical physics, such as Newtonian mechanics and electromagnetism, failed to explain key experimental observations like:

• Blackbody Radiation: Classical physics predicted that the intensity of radiation emitted by a hot object would become infinite at short wavelengths (the ultraviolet catastrophe), but experimental results showed a finite intensity distribution, which was explained by Planck’s quantized energy levels.

• Photoelectric Effect: Classical wave theory predicted that light intensity would determine the emission of electrons from a metal surface, but experiments showed that only light above a certain frequency could release electrons, regardless of intensity. This led Einstein to propose that light consists of quantized packets of energy called photons.

• Atomic Spectra: Classical theories could not explain the discrete spectra emitted by atoms. Quantum mechanics, through models like Bohr’s atom, explained that electrons exist in quantized energy levels.

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2. Photoelectric Effect

The photoelectric effect involves shining light onto a metal surface and observing the emission of electrons. Classical theory predicted that increasing light intensity would increase the energy of emitted electrons, but experiments showed:

• Threshold Frequency: Electrons were emitted only if the light’s frequency exceeded a certain threshold, regardless of light intensity.

• Instantaneous Emission: Electrons were emitted immediately when illuminated by light above the threshold frequency.

Einstein's Explanation (1905): Einstein proposed that light is quantized into photons, each with energy $E = h \nu$ where $h$ is Planck's constant and $nu$ is the frequency. If the photon energy is greater than the work function $\phi$ of the metal, an electron is ejected. This explanation clarified why light behaves as both a wave (in phenomena like interference) and as a particle (in the photoelectric effect).

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3. Concept of Matter Waves (De Broglie Waves)

In 1924, Louis de Broglie proposed that particles, like electrons, also have a wave-like nature. The de Broglie wavelength \$lambda$ of a particle is given by:

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

where $h$ is Planck’s constant and $p$ is the particle’s momentum. This equation implies that particles exhibit wave-like behavior and can undergo phenomena like diffraction and interference.

Example:

For an electron with a momentum $p = mv$ (where $m$ is the mass and $v$ is the velocity), the de Broglie wavelength becomes:

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

If an electron is moving at a speed of $10^6 \, \text{m/s}$ and has a mass $9.11 \times 10^{-31} \, \text{kg}$, its de Broglie wavelength would be:

$$\lambda = \frac{6.626 \times 10^{-34} \, \text{J·s}}{(9.11 \times 10^{-31} \, \text{kg})(10^6 \, \text{m/s})} = 7.27 \times 10^{-11} \, \text{m}$$

This wavelength is comparable to atomic dimensions and explains phenomena like electron diffraction.

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4. Wavelength of Matter Waves in Different Forms

The de Broglie wavelength depends on the momentum of the particle. For large momentum (macroscopic objects), the wavelength becomes incredibly small and unmeasurable, while for microscopic particles, the wavelength is more significant.

Example:

• Electron: For an electron with mass $m = 9.11 \times 10^{-31} \, \text{kg}$ and velocity $v = 10^6 \, \text{m/s}$, the wavelength is $\text{m}$, in the range of atomic scales, which allows for electron diffraction in microscopes.

• Baseball: A baseball with mass $m = 0.145 \, \text{kg}$ moving at $v = 20 \, \text{m/s}$ would have an incredibly small wavelength:

  $$\lambda = \frac{6.626 \times 10^{-34} \, \text{J·s}}{(0.145 \, \text{kg})(20 \, \text{m/s})} = 2.29 \times 10^{-34} \, \text{m}$$

  This wavelength is so small that it’s effectively undetectable and irrelevant for macroscopic objects.

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5. Heisenberg's Uncertainty Principle

Werner Heisenberg’s uncertainty principle (1927) states that it is impossible to simultaneously measure both the position and momentum of a particle with perfect accuracy. This is mathematically 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 $\hbar$ is the reduced Planck's constant.

Example:

For an electron confined to a box with uncertainty in position $\Delta x \approx 10^{-10} \, \text{m}$(about the size of an atom), the uncertainty in momentum would be:

$$\Delta p \geq \frac{\hbar}{2 \Delta x} = \frac{1.055 \times 10^{-34}}{2 \times 10^{-10}} \approx 5.275 \times 10^{-25} \, \text{kg·m/s}$$

This uncertainty implies that the particle's momentum (and thus velocity) cannot be known precisely if the position is confined to such a small region.

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6. Phase Velocity and Group Velocity

• Phase Velocity: The phase velocity $v_{\text{phase}}$ is the velocity at which individual wavefronts of a wave propagate. It is given by:

  $$v_{\text{phase}} = \frac{\omega}{k}$$

  where $omega$ is the angular frequency and $k$ is the wave number.

  For a matter wave (de Broglie wave), the phase velocity may be greater than the speed of light in some cases, but this does not violate relativity because no information or energy is transmitted at this velocity.

Example:

For a de Broglie wave of an electron, the phase velocity $v_{\text{phase}}$ might be calculated for specific conditions, but it’s generally less relevant than the group velocity in determining the particle’s actual speed.

• Group Velocity: The group velocity $v_{\text{group}}$ is the velocity at which the overall envelope of a wave packet propagates, and it is typically associated with the velocity of the particle itself: $$v_{\text{group}} = \frac{d\omega}{dk}$$

Example:

For an electron in a wave packet, the group velocity corresponds to the particle’s actual velocity, while the phase velocity describes the propagation of the individual wave components that make up the wave packet.

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7. Wave Function and Schrödinger’s Equation

The wave function $\psi(x,t)$ is a mathematical description of a quantum system. It contains all the information about the system, but its square modulus $|\psi(x,t)|^2$ gives the probability of finding the particle at position $x$ at time $t$.

• Schrödinger's Equation: The time-dependent Schrödinger equation governs the evolution of the wave function: $$i\hbar \frac{\partial \psi(x,t)}{\partial t} = \hat{H} \psi(x,t)$$ where $\hat{H}$ is the Hamiltonian operator (total energy), $\hbar$ is the reduced Planck’s constant, and $\psi(x,t)$is the wave function.

Example:

Consider a particle in a potential $V(x)$, with the Hamiltonian $\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)$. For a particle in a box, the solution to the time-independent Schrödinger equation gives discrete energy levels, where the wave function solutions are sinusoidal and normalized.

• Time-independent Schrödinger Equation: For systems with time-independent potentials, the equation simplifies to: $$\hat{H} \psi(x) = E \psi(x)$$ where $E$ is the energy eigenvalue. This is used to find allowed energy levels of the system.

Example:

For a particle in a box with length $L$, the allowed energy levels are:

$$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}, \quad n = 1, 2, 3, \dots$$

The wave functions corresponding to these energies are:

$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin \left(\frac{n \pi x}{L}\right)$$

This demonstrates how quantum systems have discrete energy levels, as opposed to the continuous energies predicted by classical mechanics.

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Summary

Quantum mechanics describes particles as having both wave-like and particle-like properties, and phenomena like the photoelectric effect and matter waves illustrate this duality. The uncertainty principle highlights fundamental limits to measurement, while Schrödinger’s equation governs the evolution of quantum states. Phase and group velocities describe different aspects of wave propagation in quantum systems. Together, these concepts form the core framework of quantum theory, explaining phenomena beyond the reach of classical mechanics.