Quantum Mechanics Study Guide: From Matter Waves to Tunneling

Foundations of Quantum Mechanics and the Transition from Classical Physics

Classical Mechanics (CM) is considered an approximation of Quantum Mechanics (QM). While Quantum Mechanics provides the actual results for physical systems, Classical Mechanics essentially provides the most probable result. The behavior of subatomic particles, such as electrons, protons, and neutrons, differs significantly from the macroscopic world as we know it. In the macroscopic world, knowing the present state of a system allows us to measure any quantity for the future with certainty. However, in the atomic world, values cannot be measured exactly due to the Heigenburg Uncertainty Principle. Consequently, it is impossible to understand the behavior of subatomic particles using the principles of Classical Mechanics; instead, Quantum Mechanics must be applied to the atomic world.

The Wave Function and Probability Density

According to the de Broglie hypothesis, subatomic particles can be considered as waves, termed matter waves. For mathematical analysis, these are represented by the wave function $\Psi(\text{psi})$. For a traveling mechanical wave, we consider a wave function $\text{y}(x,t)$; similarly, for a matter wave, we use $\Psi(x, y, z, t)$, which is a function of both position and time. This function can have positive or negative values. While $\Psi$ itself does not have a direct physical meaning, the square of its absolute magnitude, $|\Psi|^2$, always yields a positive value and represents the probability of finding the particle at a specific position $(x, y, z)$ at a specific time $t$. This quantity, $|\Psi|^2$, is known as the probability function or probability density. Evaluation of the probability density at a particular place and time is proportional to the probability of finding the body at that time.

Wave functions are typically complex, containing both real and imaginary parts. If a wave function is defined as $\Psi = A + iB$, its complex conjugate is $\Psi^{*} = A - iB$, where $i = \sqrt{-1}$. The product of these two is given by:

$\Psi^{*}\Psi = |\Psi|^2 = A^2 - i^2B^2 = A^2 + B^2$

Because a particle must exist somewhere at all times, the wave function must be normalized. Normalization is achieved by ensuring that the integral of the probability density over all space is equal to unity:

$\int_{-\infty}^{\infty} |\Psi|^2 dV = 1$

A wave function that satisfies this equation is described as normalized. Any acceptable wave function can be normalized by multiplying it by an appropriate constant.

Criteria for Well-Behaved Wave Functions

In order to be considered physically acceptable or "well-behaved," a wave function $\Psi$ must satisfy several conditions. First, $\Psi$ must be continuous and single-valued everywhere. Second, the partial derivatives $\frac{\partial \Psi}{\partial x}$, $\frac{\partial \Psi}{\partial y}$, and $\frac{\partial \Psi}{\partial z}$ must also be continuous and single-valued everywhere. Third, $\Psi$ must be normalizable, which implies that it must approach zero as $x \rightarrow \pm \infty$, $y \rightarrow \pm \infty$, and $z \rightarrow \pm \infty$ so that the integral of $|\Psi|^2 dV$ over all space is a finite constant. It is noted that these rules are not always strictly obeyed in model situations that only approximate reality. For example, the wave functions of a particle in a box with infinitely hard walls do not have continuous derivatives at the walls because the potential $V$ changes abruptly to zero outside the box. However, in the real world, walls are never infinitely hard, and there are no such sharp changes, meaning derivatives remain continuous.

Wave Function and Schrödinger Equation for a Free Particle

A free particle is defined as a particle not influenced by any forces, thus moving in a straight path at a constant velocity. If a free particle moves freely with a constant speed in the $x$ direction, its wave function is expressed as $\Psi = Ae^{-i\omega(t - x/v)}$. By substituting $\omega = 2\pi f$ and $v = \lambda f$, we obtain:

$\Psi = Ae^{-i2\pi f(t - x/\lambda f)} = Ae^{(-i/\hbar)[(2\pi\hbar f)t - (2\pi\hbar/\lambda)x]}$

Using the relations $\hbar = \frac{h}{2\pi}$, $E = hf = 2π\hbar f$, and $p = \frac{h}{\lambda} = \frac{2\pi\hbar}{\lambda}$, the expression becomes:

$\Psi = Ae^{(-i/\hbar)(Et - px)}$

This equation describes the wave equivalent of an unrestricted particle with total energy $E$ and momentum $p$ moving in the $+x$ direction. By differentiating $\Psi$ twice with respect to $x$, we find:

$\frac{\partial^2\Psi}{\partial x^2} = -\frac{p^2}{\hbar^2}\Psi$

$p^2\Psi = -\hbar^2 \frac{\partial^2\Psi}{\partial x^2}$

Differentiating $\Psi$ with respect to $t$ yields:

$\frac{\partial\Psi}{\partial t} = -\frac{iE}{\hbar}\Psi$

$E\Psi = i\hbar \frac{\partial \Psi}{\partial t}$

Total energy is the sum of kinetic energy (expressed as $\frac{p^2}{2m}$) and potential energy ($U(x, t)$), where $U$ represents the influence of the rest of the universe on the particle:

$E = \frac{p^2}{2m} + U(x, t)$

Multiplying by $\Psi$ gives the time-dependent one-dimensional Schrödinger equation:

$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + U\Psi$

In three dimensions, where potential energy is a function of $x, y, z$, and $t$, the equation is:

$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} (\frac{\partial^2 \Psi}{\partial x^2} + \frac{\partial^2 \Psi}{\partial y^2} + \frac{\partial^2 \Psi}{\partial z^2}) + U\Psi$

The Steady-State Schrödinger Equation

For systems where forces are independent of time, the time-dependent component of the wave function takes a specific form. The one-dimensional wave function of an unrestricted particle can be written as:

$\Psi = Ae^{-(i/\hbar)(Et - px)} = Ae^{-(iE/\hbar)t}e^{+(ip/\hbar)x} = \psi e^{-(iE/\hbar)t}$

Here, $\Psi$ is the product of a time-dependent function $e^{-(iE/\hbar)t}$ and a position-dependent function $\psi$. Substituting this into the time-dependent Schrödinger equation and dividing by the common exponential factor leads to the steady-state Schrödinger equation in one dimension:

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

In three dimensions, the steady-state equation is expressed as:

$\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} + \frac{2m}{\hbar^2}(E - U)\psi = 0$

A critical property of this equation is that solutions exist for specific values of energy $E$, meaning energy quantization is a natural element of wave mechanics and a universal characteristic of all stable systems. This can be represented in operator notation as $H\psi = E\psi$ or $\hat{H}\psi = \hat{E}\psi$.

Operators, Expectation Values, and Eigenvalues

An operator dictates a specific mathematical operation to be carried out on the quantity following it, usually denoted with a cap (e.g., $\hat{p}$ for the momentum operator).

The expectation value of a quantity represents the average value one would expect from a large number of measurements. For position, it is the location where the probability of finding the particle is highest:

$\langle x \rangle = \int_{-\infty}^{\infty} x|\Psi|^2 dx$

Similar definitions apply to expectation values for momentum and energy.

Eigenvalues and eigenfunctions arise from the steady-state Schrödinger equation. The specific energy values $E_n$ for which the equation can be solved are called eigenvalues, and the corresponding wave functions $\psi_n$ are eigenfunctions (derived from the German "Eigenwert" meaning proper value and "Eigenfunktion" meaning proper function). An eigenvalue equation is written as:

$\hat{G}u_n = G_n u_n$

In this expression, $\hat{G}$ is the operator corresponding to $G$, and each $G_n$ is a real number.

Case Study: Particle in a Box (Infinite Potential Well)

Consider a particle of mass $m$ traveling along the $x$-axis between $x = 0$ and $x = L$. The potential energy $U$ is infinite outside the box ($x \le 0$ and $x \ge L$) and assumed to be zero inside the box. Therefore, $\psi = 0$ for $x \le 0$ and $x \ge L$. The time-independent Schrödinger equation for the interior of the box is:

$\frac{d^2 \psi}{dx^2} + \frac{2m}{\hbar^2}E\psi = 0$

The general solution is $\psi = A\sin(\frac{\sqrt{2mE}}{\hbar}x) + B\cos(\frac{\sqrt{2mE}}{\hbar}x)$. Applying boundary conditions at $x = 0$ leads to $B = 0$. Applying the boundary condition at $x = L$, where $\psi = 0$, requires:

$\frac{\sqrt{2mE}}{\hbar}L = n\pi$

Where $n = 1, 2, 3, \dots$. Solving for energy gives the quantized levels:

$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$

The corresponding quantized wave functions are $\psi_n = A\sin(\frac{n\pi x}{L})$. Applying the normalization condition $\int_0^{L} \psi^2 dx = 1$ via the identity $\sin^2(\theta) = \frac{1}{2}(1 - \cos(2\theta))$ results in the normalization constant $A = \sqrt{\frac{2}{L}}$. Thus, the normalized wave function is:

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

For these states, the expectation value of position $\langle x \rangle$ is always $L/2$, regardless of the quantum number $n$. This reflects the symmetry of $|\psi|^2$ about the middle of the box. However, the expectation value of momentum $\langle p \rangle$ is $0$. This is because the particle moves back and forth with momentum eigenvalues $p_n = \pm \frac{n\pi\hbar}{L}$, and the average of these opposing values is zero.

Finite Potential Wells and the Tunnel Effect

In reality, potential energies are never infinite. In a finite potential well of height $U$ and width $L$, a particle with energy E < U can penetrate the classical forbidden regions (regions I and III). Classically, the particle would reflect perfectly, but quantum mechanically, there is a finite probability of the particle being found outside the walls.

This leads to the tunnel effect (tunneling), where a particle approaches a potential barrier and has a finite chance of penetrating even if its kinetic energy is less than the barrier height. This occurs notably in alpha radiation, where an alpha particle with a few $MeV$ of energy escapes a nucleus with a potential wall of approximately $25\,MeV$. Tunneling is also essential to semiconductor diodes. The approximate transmission probability $T$ is:

$T = e^{-2k_2L}$

Where $k_2 = \frac{\sqrt{2m(U - E)}}{\hbar}$ and $L$ is the barrier width. A practical application is the Scanning Tunneling Microscope (STM), which uses a piezoelectric tube and distance control units to measure tunneling current between a tip and a sample surface.

Worked Examples from the Course

Example 5.2: A particle on the $x$-axis has $\psi = ax$ between $x=0$ and $x=1$, and $0$ elsewhere. (a) Probability between $x=0.45$ and $x=0.55$: $P = \int_{0.45}^{0.55} |\psi|^2 dx = a^2 \int_{0.45}^{0.55} x^2 dx = a^2 [\frac{x^3}{3}]_{0.45}^{0.55} = 0.0251a^2.$ (b) Expectation value $\langle x \rangle$: $\langle x \rangle = \int_0^1 x|\psi|^2 dx = a^2 \int_0^1 x^3 dx = a^2 [\frac{x^4}{4}]_0^1 = \frac{a^2}{4}.$

Example 5.3: Find the eigenvalue of the operator $\frac{d^2}{dx^2}$ for the eigenfunction $\psi = e^{2x}$. Applying the operator: $\frac{d^2}{dx^2}(e^{2x}) = \frac{d}{dx}(2e^{2x}) = 4e^{2x}$. Since $\hat{G}\psi = 4\psi$, the eigenvalue is $4$.

Example 5.4: Probability for a particle in a box to be between $0.45L$ and $0.55L$. For $n=1$ (ground state), the probability is $19.8\%$. For $n=2$ (first excited state), the probability is $0.65\%$. The classical expectation is $10\%$.

Example 5.6: Electrons with $1.0\,eV$ and $2.0\,eV$ incident on a $10.0\,eV$ barrier, $0.50\,nm$ wide. For $1.0\,eV$: $k_2 = 1.6 \times 10^{10}\,m^{-1}$. $2k_2L = 16$. $T_1 = e^{-16} = 1.1 \times 10^{-7}$. For $2.0\,eV$, the probability $T_2 = 2.4 \times 10^{-7}$. Doubling the width to $1.0\,nm$ reduces probabilities to $T_1 = 1.3 \times 10^{-14}$ and $T_2 = 5.1 \times 10^{-14}$, showing that $T$ is more sensitive to width than energy.

Questions & Discussion

Practice Problem 3: Which of the following wave functions cannot be solutions of Schrödinger's equation for all values of $x$? (a) $\psi = A\sec(x)$; (b) $\psi = A\tan(x)$; (c) $\psi = Ae^{x^2}$; (d) $\psi = Ae^{-x^2}$. Answer Criteria: Functions that go to infinity ($\sec$, $\tan$ at certain points, and $e^{x^2}$ as $x \rightarrow \infty$) are not well-behaved.

Practice Problem 4: Find the normalization constant $A$ for $\psi = Axe^{-x^2/2}$.

Practice Problem 5: $\psi = A\cos^2(x)$ for -\pi/2 < x < \pi/2. Find $A$ and the probability between $x=0$ and $x=\pi/4$.

Practice Problem 10: Find eigenvalues for $\frac{d^2}{dx^2}$ with eigenfunction $\sin(nx)$.

Practice Problem 19: Find probability for a particle in a box to be between $x=0$ and $x=L/n$ in the $n$-th state.

Practice Problem 24: Find approximate probability for electrons with $0.400\,eV$ to penetrate a $3.00\,eV$ barrier that is $0.100\,nm$ wide.

Practice Problem 25: Find the energy required for $1.00\%$ of electrons to penetrate a $6.00\,eV$ barrier, $0.200\,nm$ wide.

Wave Function, Probability Density, and Normalization

The wave function, denoted as $\Psi$, represents the state of a quantum particle. It contains all necessary information about the particle's behavior and is a mathematical function that depends on position and time. To find the likelihood of locating this particle at a specific position, we calculate the probability density, which is the square of the absolute value of the wave function: $|\Psi|^2$. This ensures that the total probability of finding the particle anywhere in space equals one, a condition known as normalization. The normalization condition is expressed mathematically as: $\int_{-\infty}^{\infty} |\Psi|^2 dV = 1$.

Expectation Value of x

The expectation value, $\langle x \rangle$, is the average position where we expect to find the particle. It is calculated using the formula: $\langle x \rangle = \int_{-\infty}^{\infty} x |\Psi|^2 dx$. This integral helps us determine the particle's average location based on its wave function.

Criteria for Well-Behaved Wave Functions

For a wave function to be considered physically acceptable, it must fulfill several criteria:

• It should be continuous and single-valued everywhere, meaning it does not jump to different values without reason.

• The derivatives (which measure how the function changes) with respect to position should also be continuous and single-valued.

• It must be normalizable, which means it should approach zero as position goes to infinity. This ensures that the total probability remains finite.

Operators, Eigenfunctions, and Eigenvalues

An operator, typically represented with a hat (e.g., $\hat{G}$), is a mathematical rule that acts on a wave function to yield another function. Eigenfunctions are special types of wave functions that satisfy the equation $\hat{G}\psi_n = G_n \psi_n$, where the values $G_n$ are referred to as eigenvalues. These eigenvalues signify specific measurable quantities associated with the quantum state.

Schrödinger Equation for Free Particles

For a free particle moving without any forces acting on it, the time-dependent Schrödinger equation is: $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2}$.

The time-independent version of the Schrödinger equation for the same particle is: $-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + U\psi = E\psi$, where $E$ represents the particle's energy.

Schrödinger Equation for a Particle in a One-Dimensional Box

For a particle confined in a box between $x=0$ and $x=L$, where the walls are considered infinitely high (a concept called infinite potential), the time-independent Schrödinger equation leads to quantized energy levels: $E_n = \frac{n^2\pi^2 \hbar^2}{2mL^2}$, with $n = 1, 2, 3, \ldots$. The normalized wave functions can be expressed as: $\psi_n = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$.

Wave Functions and Probability Density Representation

Graphically, the wave functions demonstrate sinusoidal patterns based on the quantum number $n$, with their corresponding probability densities showing peaks that indicate where the particle is likely found. If the potential walls became finite (as in a finite potential well), the wave function would not abruptly drop to zero outside the box, but instead, would show a smooth decrease, indicating some probability of the particle being outside the box.

Tunnel Effect and Examples

The tunnel effect is a phenomenon where particles can pass through potential barriers, even if they do not appear to have enough energy to do so. For instance, in alpha decay, alpha particles can tunnel out of an atomic nucleus. In Scanning Tunneling Microscopes (STM), tunneling is used to measure the surface of materials at the atomic level by detecting current flow between a conductive tip and a surface. In diodes, electrons can tunnel through barriers at junctions, allowing current to flow even when they do not have enough energy to overcome the barrier directly.

Transmission Probability

Transmission probability refers to the chance that a particle will successfully tunnel through a barrier and is expressed mathematically as: $T = e^{-2k_2L}$, where:

• $k_2 = \frac{\sqrt{2m(U-E)}}{\hbar}$

• $L$ is the width of the barrier.

This formula shows how easier tunneling becomes as the width $L$ of the barrier decreases or as the energy $E$ of the particle approaches the height of the barrier. Each term in the expression plays a significant role in determining the likelihood of tunneling occurring.

In quantum mechanics, the behavior of a free particle can be described using the Schrödinger equation. This can be expressed in both time-dependent and time-independent forms.

Time-Dependent Schrödinger Equation

To derive the time-dependent Schrödinger equation for a free particle, we start from the fundamental principle that the state of the system can be described by a wave function $\Psi(x, t)$:

1. According to De Broglie, a particle can be associated with a wave. Therefore, we can represent the wave function as a plane wave:    $\Psi(x, t) = Ae^{-i(\omega t - kx)}$    where:    - $A$ is the amplitude,    - $\omega = 2\pi f$ is the angular frequency,    - $k = \frac{2\pi}{\lambda}$ is the wave number, and    - $\lambda$ is the wavelength of the wave.

2. The energy of the particle is connected to its frequency by:    $E = h f = \hbar \omega$    where $\hbar$ is the reduced Planck's constant.

3. The momentum of the particle relates to its wave number by:    $p = \hbar k$

4. Substituting these relations into the wave function, we get:    $\Psi(x, t) = A e^{-\frac{i}{\hbar}(E t - p x)}$

5. Now we need to calculate the time and spatial derivatives of the wave function:    - The first derivative with respect to time:    $\frac{\partial \Psi}{\partial t} = -\frac{iE}{\hbar} \Psi$    - The second derivative with respect to position:    $\frac{\partial^2 \Psi}{\partial x^2} = -\frac{p^2}{\hbar^2} \Psi$

6. Substituting these derivatives into the equation:    $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2}$

This leads us to the time-dependent Schrödinger equation:    $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2}$

Time-Independent Schrödinger Equation

To derive the time-independent Schrödinger equation, we assume a separable solution of the form:    $\Psi(x, t) = \psi(x) e^{-i\frac{E}{\hbar}t}$

1. Substituting this form into the time-dependent Schrödinger equation, we have:    $i\hbar \left( -i\frac{E}{\hbar} \psi(x)e^{-i\frac{E}{\hbar}t} \right) = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x)}{\partial x^2} e^{-i\frac{E}{\hbar}t}$

2. Canceling the exponential terms from both sides gives us:    $E\psi(x) = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x)}{\partial x^2}$

3. Rearranging terms, we achieve the time-independent Schrödinger equation:    $-\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x)}{\partial x^2} + U\psi(x) = E\psi(x)$    where $U$ is the potential energy (for a free particle, $U = 0$).

Thus, the time-independent Schrödinger equation simplifies to the following form for a free particle:    $-\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x)}{\partial x^2} = E\psi(x)$.

Summary

In summary, the time-dependent Schrödinger equation describes how a quantum state evolves in time, while the time-independent equation allows us to analyze stationary states and their energy levels, crucial for understanding the behavior of quantum systems.

Deriving sc eqn for 1d particle

To derive the Schrödinger equation for a particle confined in a one-dimensional box with infinitely high walls, we begin with the time-independent Schrödinger equation. This equation is expressed as:

$- rac{\hbar^2}{2m} rac{d^2 \psi(x)}{dx^2} + U(x) \psi(x) = E \psi(x)$

−2mℏ2​dx2d2ψ(x)​+U(x)ψ(x)=Eψ(x)

Where:

• $\hbar$ is the reduced Planck's constant.

• $m$ is the mass of the particle.

• $U(x)$ is the potential energy.

• $E$ is the total energy of the particle.

In the case of a particle in a box, we assume that the walls of the box are infinitely high. This implies that the potential energy outside the box is infinitely large, while the potential energy inside the box is zero. Therefore, we can write:

U(x) = 0 \quad \text{for} \quad 0 < x < L
$U(x) = \infty \quad \text{for} \quad x \leq 0 \text{ or } x \geq L$

Hence, the time-independent Schrödinger equation for the region inside the box (where $U(x) = 0$) simplifies to:

$- rac{\hbar^2}{2m} rac{d^2 \psi(x)}{dx^2} = E \psi(x)$

−2mℏ2​dx2d2ψ(x)​=Eψ(x)

Rearranging this gives:

$\frac{d^2 \psi(x)}{dx^2} + \frac{2mE}{\hbar^2} \psi(x) = 0$

Let’s define a new constant:

$k^2 = \frac{2mE}{\hbar^2}$

Thus, we rewrite the equation as:

$\frac{d^2 \psi(x)}{dx^2} + k^2 \psi(x) = 0$

The general solution to this second-order differential equation is:

$\psi(x) = A \sin(kx) + B \cos(kx)$

Where $A$ and $B$ are constants to be determined by the boundary conditions.

Boundary Conditions

Since the walls are infinitely high, the wave function must satisfy the following boundary conditions:

1. $\psi(0) = 0$ (the wave function must be zero at the left wall)

2. $\psi(L) = 0$ (the wave function must also be zero at the right wall)

Applying the first boundary condition at $x = 0$:

$\psi(0) = A \sin(0) + B \cos(0) = B = 0$

This implies that $B = 0$, so the wave function simplifies to:

$\psi(x) = A \sin(kx)$

Next, we apply the second boundary condition at $x = L$:

$\psi(L) = A \sin(kL) = 0$

For this equation to hold, either $A = 0$ (which is not physically meaningful since the wave function must not be identically zero) or:

$\sin(kL) = 0$

This leads us to:

$kL = n\pi$ Where $n = 1, 2, 3, ext{…}$ (positive integers).

Thus, we can express $k$ as:

$k = \frac{n\pi}{L}$

Expression for Wave Functions

Substituting the expression for $k$ back into our wave function yields:

$\psi(x) = A \sin\left(\frac{n\pi}{L}x\right)$

Normalization

To ensure the wave function is normalized, we need to find the constant $A$ such that:

$\int_0^L |\psi(x)|^2 dx = 1$

Calculating this integral gives us:

$\int_0^L |A \sin\left(\frac{n\pi}{L}x\right)|^2 dx = |A|^2 \int_0^L \sin^2\left(\frac{n\pi}{L}x\right) dx = |A|^2 \left(\frac{L}{2}\right) = 1$

Thus,

$|A|^2 \cdot \frac{L}{2} = 1 \Rightarrow |A|^2 = \frac{2}{L} \Rightarrow A = \sqrt{\frac{2}{L}}$

The normalized wave functions become:

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

Energy Levels

Finally, the energy levels associated with these wave functions can be found using:

$E_n = \frac{\hbar^2 k^2}{2m}$

Substituting $k = \frac{n\pi}{L}$ gives:

$E_n = \frac{\hbar^2}{2m} \left(\frac{n\pi}{L}\right)^2$

Thus, the quantized energy levels for the particle in the box are:

$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$

In conclusion, for a particle in a one-dimensional box with infinitely high walls, the normalized wave functions are given by:

$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi}{L}x\right)$
while the quantized energy levels are:

$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$.