Phy 129 - WM, CT-03

Statement: For position (

x

) and momentum (

p

), the uncertainty principle is:

\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 \approx 1.05 \times 10^{-34} \, \text{J·s}

Similarly, for energy ( $E$ ) and time ( $t$ ): $\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$ .

#Learning -2

$p = \gamma m v$ , where:

$m$ is the rest mass of the particle,
$v$ is the velocity of the particle,
$\gamma$ is the Lorentz factor, given by $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$

A wave group:

a wave group corresponds to a moving particle and is formed by the superposition of multiple harmonic waves with different frequencies, wave numbers, and amplitudes.\

Wave Function $\Psi(x)$ :

$\Psi(x)$ is the wave function describing the wave group at position $x$ at a specific time $t$ .
It represents the spatial distribution of the particle’s wave, where $|\Psi(x)|^2$ gives the probability density of finding the particle at position $x$

Fourier Integral:

The integral $\Psi(x) = \int_0^{\infty} g(k) \cos k x \, dk$ shows that $\Psi(x)$ is formed by summing (integrating) an infinite number of cosine waves ( $\cos k x$ ) with different wave numbers $k$
Each cosine wave has an amplitude determined by the function $g(k)$ .

Function $g(k)$ :

On Page 4, $g(k)$ is described as the Fourier transform of $\Psi(x)$ .
It specifies how the amplitudes of the contributing waves vary with wave number $k$ .
$g(k)$ completely describes the wave group, just as $\Psi(x)$ does, but in the wave number (or momentum) domain

Difference between psi(x) and it's square function:

Def: Ψ

\Psi(x)

is the wave function of a wave group representing a particle at position

x

at a specific time

t

|\Psi(x)|^2

is the absolute square of the wave function, representing the probability density of finding the particle at position

x

$\Psi(x)$ : A complex-valued function describing the wave-like behavior of a particle. It includes both amplitude and phase information.
$|\Psi(x)|^2$ : A real, non-negative number representing the probability density of finding the particle at position $x$

properties of the wave function ( $\psi$ ) are:

Continuity and Single-Valued: The wave function $\psi$ must be continuous and single-valued everywhere. This ensures that the wave function does not have abrupt changes or multiple values at any point in space.
Continuous and Single-Valued Derivatives: The first derivatives of the wave function with respect to spatial coordinates, $\frac{d\psi}{dx}$ , $\frac{d\psi}{dy}$ , and $\frac{d\psi}{dz}$ , must also be continuous and single-valued everywhere. This ensures smooth behavior of the wave function across space.
Normalizability: The wave function $\psi$ must be normalizable, meaning it must approach zero as the spatial coordinates $x \to \infty$ , $y \to \infty$ , and $z \to \infty$ . This ensures that the integral of the probability density $|\psi|^2$ over all space, $\int_{-\infty}^{\infty} |\psi|^2 dV$ , is a finite constant, specifically equal to 1 for a normalized wave function, indicating that the particle exists somewhere in space.

Formula: $\psi = A + iB$
- Context: Describes a complex wave function $\psi$ , where $A$ and $B$ are real functions.
- Explanation: The wave function $\psi$ is a complex number, meaning it has a real part ( $A$ ) and an imaginary part ( $iB$ , where $i = \sqrt{-1}$ ). Think of it like a mathematical way to describe a particle’s behavior, combining two parts to capture its quantum properties.
Formula: $\psi^* = A - iB$
- Context: Defines the complex conjugate $\psi^*$ of the wave function $\psi$ .
- Explanation: The complex conjugate $\psi^*$ is found by changing the sign of the imaginary part of $\psi$ . If $\psi = A + iB$ , then $\psi^* = A - iB$ . This is used to calculate probabilities because probabilities must be real numbers.
Formula: $\psi \psi^* = A^2 + B^2$
- Context: Shows how to get the probability density from $\psi$ and its conjugate $\psi - i\psi$ .
- Explanation: Multiplying $\psi$ by its conjugate $\psi^*$ gives $A^2 + B^2$ , which is always a positive real number. This value, $|\psi|^2$ , is proportional to the probability of finding the particle at a specific place and time. It’s like a measure of the “strength” of the wave function at that point.
Formula: $\int_{-\infty}^{\infty} |\psi|^2 dV = 1$
- Context: Describes the normalization condition for a wave function.
- Explanation: The total “area” under the probability density $|\psi|^2$ over all space equals 1 for a normalized wave function. This means the particle is definitely somewhere to exist somewhere in space, so the probabilities add up to 100%. The integral sums up $|\psi|^2$ over all space (volume $dV$ ).
Formula: $\int_{-\infty}^{\infty} |\psi|^2 dV = 0$
- Context: Indicates a non-existent particle.
- Explanation: If the integral of $|\psi|^2$ over all space equals 0, it means the particle doesn’t exist anywhere. There’s zero probability of finding it, so the wave function doesn’t describe a real particle.

Page 6

Formula: $\Psi = A e^{-i \omega (t - x/v)}$
- Context: Represents the wave function $\Psi$ for a particle moving freely in the $+x$ direction.
- Explanation: This describes a wave-like behavior for a free particle. Here, $A$ is the amplitude (size of the wave), $\omega$ is related to frequency, $t$ is time, $x$ is position, and $v$ is the speed of the wave. The exponent makes it a complex wave that moves along the $x$ -axis over time.
Formula: $\Psi = A e^{-2 \pi i (\nu t - x/\lambda)}$
- Context: Rewritten form of the wave function using frequency $\nu$ and wavelength $\lambda$ .
- Explanation: This is the same wave function as above, but rewritten using $\nu$ (frequency) and $\lambda$ (wavelength). It’s a more convenient form because $\nu$ and $\lambda$ relate to the particle’s energy and momentum. The wave still describes a particle moving in the $+x$ direction.
Formula: $E = h \nu = 2 \pi \hbar \nu$
- Context: Relates the particle’s energy $E$ to its frequency $\nu$ .
- Explanation: The energy $E$ of a particle is equal to Planck’s constant $h$ times the frequency $\nu$ , or equivalently $2 \pi \hbar \nu$ (where $\hbar = h / 2\pi$ ). This connects the particle’s energy to its wave-like behavior, showing how quantum mechanics links energy to waves.
Formula: $\lambda = \frac{h}{p} = \frac{2 \pi \hbar}{p}$
- Context: Relates the wavelength $\lambda$ to the particle’s momentum $p$ .
- Explanation: The wavelength $\lambda$ of the particle’s wave is Planck’s constant $h$ divided by its momentum $p$ , or $2 \pi \hbar / p$ . Momentum is mass times velocity, so this formula shows how a particle’s motion determines its wave properties.
Formula: $\Psi = A e^{-(i/\hbar)(E t - p x)}$
- Context: Wave function for a free particle with energy $E$ and momentum $p$ .
- Explanation: This is the final form of the wave function for a free particle, using energy $E$ and momentum $p$ . It describes a wave moving in the $+x$ direction, where $E t - p x$ in the exponent combines energy and momentum to define how the wave evolves over time and space.

Page 7

Formula: $\frac{\partial^2 \Psi}{\partial x^2} = -\frac{p^2}{\hbar^2} \Psi$
- Context: Result of differentiating the wave function $\Psi$ twice with respect to $x$ .
- Explanation: When you take the second derivative of the wave function $\Psi$ with respect to position $x$ , you get a result proportional to $\Psi$ itself, scaled by the momentum squared ( $p^2$ ) divided by $\hbar^2$ . This shows how the wave function curves in space, related to the particle’s momentum.
Formula: $p^2 \Psi = -\hbar^2 \frac{\partial^2 \Psi}{\partial x^2}$
- Context: Rearranged form of the previous equation.
- Explanation: This just flips the previous formula to express the momentum term $p^2 \Psi$ . It shows that the momentum squared acting on the wave function is related to its second derivative, a key step toward Schrödinger’s equation.
Formula: $\frac{\partial \Psi}{\partial t} = -\frac{i E}{\hbar} \Psi$
- Context: Result of differentiating the wave function $\Psi$ with respect to time $t$ .
- Explanation: Taking the first derivative of $\Psi$ with respect to time gives a result proportional to $\Psi$ , scaled by the energy $E$ and constants. This shows how the wave function changes over time, driven by the particle’s energy.
Formula: $E \Psi = -\frac{\hbar}{i} \frac{\partial \Psi}{\partial t}$
- Context: Rearranged form of the previous equation.
- Explanation: This rewrites the time derivative to express the energy term $E \Psi$ . It shows that the energy acting on the wave function is related to how it changes with time, another step toward Schrödinger’s equation.

Page 8

Formula: $E = \frac{p^2}{2m} + U(x, t)$
- Context: Total energy $E$ of a particle at non-relativistic speeds.
- Explanation: The total energy $E$ is the sum of kinetic energy ( $p^2 / 2m$ , where $p$ is momentum and $m$ is mass) and potential energy $U$ , which depends on position $x$ and time $t$ . This is like saying a particle’s energy comes from its motion plus its position in a force field (like gravity or electric fields).
Formula: $E \Psi = \frac{p^2 \Psi}{2m} + U \Psi$
- Context: Energy equation multiplied by the wave function $\Psi$ .
- Explanation: This applies the energy formula to the wave function. It says the total energy times $\Psi$ equals the kinetic energy term ( $p^2 \Psi / 2m$ ) plus the potential energy term ( $U \Psi$ ). It’s setting up the components for Schrödinger’s equation.
Formula: $i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + U \Psi$
- Context: Time-dependent Schrödinger equation in one dimension.
- Explanation: This is the Schrödinger equation, which describes how the wave function $\Psi$ changes over time and space. The left side ( $i \hbar \frac{\partial \Psi}{\partial t}$ ) relates to the energy and time evolution. The right side has two parts: $-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2}$ for kinetic energy (how the wave curves in space) and $U \Psi$ for potential energy (how forces affect the particle). It’s like a rule for how a particle’s wave behaves under forces.

Page 9

Formula: $i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \left( \frac{\partial^2 \Psi}{\partial x^2} + \frac{\partial^2 \Psi}{\partial y^2} + \frac{\partial^2 \Psi}{\partial z^2} \right) + U \Psi$
- Context: Time-dependent Schrödinger equation in three dimensions.
- Explanation: This is the 3D version of the Schrödinger equation. It’s the same idea as the 1D version but accounts for motion in $x$ , $y$ , and $z$ directions. The term $\frac{\partial^2 \Psi}{\partial x^2} + \frac{\partial^2 \Psi}{\partial y^2} + \frac{\partial^2 \Psi}{\partial z^2}$ describes how the wave function curves in all three dimensions (like 3D kinetic energy). $U \Psi$ still represents potential energy, which can vary with $x$ , $y$ , $z$ , and time $t$ . It predicts the particle’s behavior in a 3D space with forces.

Page 12

Formula: $\Psi = a_1 \Psi_1 + a_2 \Psi_2$
- Context: Superposition principle for wave functions.
- Explanation: If $\Psi_1$ and $\Psi_2$ are two valid wave functions (solutions to Schrödinger’s equation), you can combine them by multiplying each by constants ( $a_1$ , $a_2$ ) and adding them to get a new valid wave function $\Psi$ . This is called superposition, meaning quantum waves can add up like regular waves (e.g., light or sound), leading to effects like interference.

Page 13

Formula: $P_1 = |\Psi_1|^2 = \Psi_1^* \Psi_1$
- Context: Probability density for electrons passing through slit 1 in a double-slit experiment.
- Explanation: The probability $P_1$ of finding an electron at the screen when only slit 1 is open is given by $|\Psi_1|^2$ , which is $\Psi_1$ times its complex conjugate $\Psi_1^*$ . This gives the likelihood of detecting the electron at a specific spot, based on the wave function $\Psi_1$ for slit 1.

Page 14

Formula: $P_2 = |\Psi_2|^2 = \Psi_2^* \Psi_2$
- Context: Probability density for electrons passing through slit 2 in a double-slit experiment.
- Explanation: Similar to the previous formula, $P_2$ is the probability of finding an electron when only slit 2 is open. It’s calculated as $|\Psi_2|^2$ , or $\Psi_2$ times its conjugate $\Psi_2^*$ , based on the wave function $\Psi_2$ for slit 2.
Formula: $\Psi = \Psi_1 + \Psi_2$
- Context: Superposition of wave functions in the double-slit experiment with both slits open.
- Explanation: When both slits are open, the total wave function $\Psi$ is the sum of the wave functions $\Psi_1$ (from slit 1) and $\Psi_2$ (from slit 2). This adding of wave functions creates the interference pattern seen on the screen.
Formula: $P = |\Psi|^2 = |\Psi_1 + \Psi_2|^2 = (\Psi_1^* + \Psi_2^*)(\Psi_1 + \Psi_2) = \Psi_1^* \Psi_1 + \Psi_2^* \Psi_2 + \Psi_1^* \Psi_2 + \Psi_2^* \Psi_1$
- Context: Probability density for the double-slit experiment with both slits open.
- Explanation: The probability $P$ of finding an electron on the screen when both slits are open is $|\Psi|^2$ , where $\Psi = \Psi_1 + \Psi_2$ . Expanding this gives $P_1 = |\Psi_1|^2$ , $P_2 = |\Psi_2|^2$ , plus two extra terms ( $\Psi_1^* \Psi_2 + \Psi_2^* \Psi_1$ ). These extra terms cause the interference pattern (alternating bright and dark spots), showing that wave functions add, not probabilities.
Formula: $P = P_1 + P_2 + \Psi_1^* \Psi_2 + \Psi_2^* \Psi_1$

Context: Simplified form of the previous probability density.
Explanation: This rewrites the probability $P$ as the sum of individual probabilities $P_1$ and $P_2$ (for each slit alone) plus the interference terms $\Psi_1^* \Psi_2 + \Psi_2^* \Psi_1$ . The interference terms are why the pattern with both slits open differs from just adding the single-slit patterns.

Example 1: Electron Orbit Radius in a Hydrogen Atom (Page 2)

Context: Comparison between Bohr’s theory and quantum mechanics regarding the electron’s orbit in a ground-state hydrogen atom.
Description:
- Bohr’s theory claims the electron’s orbit radius in a ground-state hydrogen atom is exactly $5.3 \times 10^{-11} \, \text{m}$ .
- Quantum mechanics, however, describes this radius in terms of probabilities.
Answer (Outcome):
- Quantum mechanics states that $5.3 \times 10^{-11} \, \text{m}$ is the most probable radius for the electron in the ground-state hydrogen atom.
- In experiments, most trials will yield different values (larger or smaller), but the value most likely to be found is $5.3 \times 10^{-11} \, \text{m}$ .
- This highlights quantum mechanics’ focus on probabilities rather than definite values, unlike Bohr’s precise radius.

Example 2: Double-Slit Experiment with Electrons (Pages 13–14)

Context: Illustration of the superposition principle using a double-slit experiment with electrons, as shown in Figure 5.2.
Description:
- A parallel beam of monoenergetic electrons passes through two slits (slit 1 and slit 2) toward a viewing screen.
- The experiment considers three cases:
  - Only slit 1 open (Figure 5.2b).
  - Only slit 2 open (Figure 5.2c).
  - Both slits open (Figure 5.2e).
- The wave functions $\Psi_1$ (for slit 1) and $\Psi_2$ (for slit 2) are used to calculate probability densities $P = |\Psi|^2$ .
Answer (Outcome):
- Slit 1 Only Open (Figure 5.2b):
  - The electron intensity at the screen corresponds to the probability density $P_1 = |\Psi_1|^2 = \Psi_1^* \Psi_1$ .
  - This produces a single-slit diffraction pattern.
- Slit 2 Only Open (Figure 5.2c):
  - The electron intensity corresponds to $P_2 = |\Psi_2|^2 = \Psi_2^* \Psi_2$ .
  - This also produces a single-slit diffraction pattern, similar but shifted due to the different slit position.
- Both Slits Open (Figure 5.2e):
  - The total wave function is $\Psi = \Psi_1 + \Psi_2$ .
  - The probability density is $P = |\Psi|^2 = |\Psi_1 + \Psi_2|^2 = \Psi_1^* \Psi_1 + \Psi_2^* \Psi_2 + \Psi_1^* \Psi_2 + \Psi_2^* \Psi_1$ .
  - This results in an interference pattern with alternating maxima and minima, similar to light passing through a double slit (as in Figure 2.4).
  - The pattern is not the sum of individual intensities $P_1 + P_2$ (Figure 5.2d), because wave functions add, not probabilities.
  - The extra terms $\Psi_1^* \Psi_2 + \Psi_2^* \Psi_1$ cause the oscillations in intensity, demonstrating interference.
- Key Insight: The interference pattern (Figure 5.2e) shows that quantum mechanics involves the superposition of wave functions, leading to wave-like behavior for electrons, unlike classical particles.

Example 3: Applications of Schrödinger’s Equation (Page 15)

Context: Practical applications where Schrödinger’s equation is used to model quantum mechanical systems.
Description: The PDF lists several technological applications where solving Schrödinger’s equation helps understand or design systems involving quantum effects.
Answer (Outcome):

Transistors and Integrated Circuits (ICs):
- Schrödinger’s equation is used to model electron behavior in quantum wells, barriers, and heterostructures in transistors.
- It helps understand how electrons move through materials, especially in nano-sized transistors (e.g., in advanced CPUs), where quantum effects are significant.
Quantum Tunneling:
- Devices like tunnel diodes and tunneling field-effect transistors (TFETs) rely on quantum tunneling.
- Solving Schrödinger’s equation predicts how electrons can “tunnel” through barriers, enabling these devices’ functionality.
Quantum Computing and Information Processing:
- Quantum computers use qubits, which follow quantum mechanical principles.
- Schrödinger’s equation models how qubits evolve over time, which is crucial for designing quantum logic gates and algorithms.
Mechanical and Aerospace Engineering:
- In Micro-Electro-Mechanical Systems (MEMS) and Nano-Electro-Mechanical Systems (NEMS), quantum effects like energy quantization are important.
- Schrödinger’s equation is used to model these effects for designing nanoscale sensors and actuators.
Key Insight: Schrödinger’s equation is essential for predicting and designing systems where quantum mechanics governs behavior, enabling modern technologies.

Aspect	Quantum Wave Function	Classical Wave Function
Nature	Probabilistic, describes quantum state	Physical, describes measurable disturbance
Observable	$$	\psi
Value	Complex-valued	Typically real-valued
Equation	Schrödinger equation	Classical wave equation
Collapse	Collapses upon measurement	No collapse, evolves deterministically
Uncertainty	Leads to Heisenberg Uncertainty Principle	No quantum uncertainty
Space	Configuration/Hilbert space	Physical 3D space
Examples	Electron in atom, quantum harmonic oscillator	Sound wave, electromagnetic wave, water wave

Feature	Classical Wave Function	Quantum Wave Function
Definition	Describes a physical disturbance (e.g., displacement, electric field).	Describes the quantum state of a particle/system.
Physical Meaning	Directly measurable (e.g., $𝑢 (𝑥, 𝑡)$ is displacement).	$$
Mathematical Nature	Typically real-valued, physical units.	Complex-valued, normalized ($$ \int
Governing Equation	Wave equation: $\frac{\partial^{2} 𝑢}{\partial 𝑡^{2}} = 𝑣^{2} \frac{\partial^{2} 𝑢}{\partial 𝑥^{2}}$ .	Schrödinger equation: $𝑖 ℏ \frac{\partial 𝜓}{\partial 𝑡} = \hat{𝐻} 𝜓$ .
Probabilistic?	Deterministic, no probabilities.	Probabilistic, governs measurement outcomes.
Uncertainty	No inherent uncertainty principle.	Leads to $Δ 𝑥 \cdot Δ 𝑝 \geq \frac{ℏ}{2}$ .
Examples	Sound wave: $𝑢 (𝑥, 𝑡) = 𝐴 \cos (𝑘 𝑥 - 𝜔 𝑡)$ .	Electron wave packet: $𝜓 (𝑥, 𝑡) = 𝐴 𝑒^{- \frac{𝑥^{2}}{4 𝜎^{2}}} 𝑒^{𝑖 𝑘_{0} 𝑥}$ .

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