Question

The general form for symmetric and antisymmetric wave functions is \(\psi_{n}\left(x_{1}\right) \psi_{n}\left(x_{2}\right) \pm \psi_{n} \cdot\left(x_{1}\right) \psi_{n}\left(x_{2}\right)\) but it is not nornalized. (a) In applying quantum mechanics, we usually deal with quantum states that are "orthonormal." That is, if we integrate over all space the square of any individual-particle function, such as \(\psi_{n}^{*}(x) \psi_{n}(x)\) or \(\psi_{i}^{\circ}(x) \psi_{n}(x),\) we get \(1,\) but for the product of different individual-particle functions, such as \(\psi_{n}^{\circ}(x) \psi_{u}(x)\), we get 0 . This happens to be true for all the systems in which we have obtained or tabulated sets of wave functions (e.g., the particle in a box, the harmonic oscillator, and the hydrogen atom). Assuming that this holds, what multiplicative constant would normalize the symmetric and antisymmetric functions? (b) What value \(A\) gives the vector \(\mathbf{V}=A(\hat{\mathbf{x}} \pm \hat{\mathbf{y}})\) unit length? (c) Discuss the relationship between your answers in (a) and (b).

Short answer

The multiplicative constant that will normalize the symmetric and antisymmetric functions is \(1 / \sqrt{2}\). For the vector V, the value of A that provides a unit length is also \(1 / \sqrt{2}\). The answers to both (a) and (b) are equivalent, illustrating the universal application of normalization procedures in different contexts.

Step-by-step solution

Normalization of Symmetric and Antisymmetric Functions

In terms of the antisymmetric function, let's normalize \(\psi_{n}(x_{1})\psi_{n}(x_{2}) - \psi_{n}(x_{2})\psi_{n}(x_{1})\) to find the required constant. Observe that the normalization condition for these functions involves a double integral over all space, which should equal 1. Solve the integral \( \int \int (\psi_{n}(x_{1})\psi_{n}(x_{2}) - \psi_{n}(x_{2})\psi_{n}(x_{1}))^{2} dx_{1} dx_{2} = 1 \) . This simplifies to \( 2(\int \psi_{n}(x_{1})^{2} dx_{1})^{2} = 1 \) by using the fact that \(\int \psi_{n}(x)^{2} dx = 1\) . Hence, the multiplicative constant would be \( 1/\sqrt{2} \) . For the symmetric function, repeat the same steps but with a positive sign. The result will be the same since the squared functions are the same for both symmetric and antisymmetric cases.

Determination of Normalization Factor for Vector V

Vector V has unit length, which implies that \( A^{2}(x^{2} + y^{2} + 2x*y) = 1 \), assuming that the vector components are normalized to 1. Considering that both vector components are normalized and orthonormal to each other, this simplifies to \( A^{2} * (1 + 1) = 1 \). Solving results in \( A = 1/\sqrt{2} \)

Comparing Results

The normalization constants found for both the wave functions and vector V are equivalent, being \( 1/\sqrt{2} \). This shows a similarity in the process of normalizing objects, be they quantum states or mathematical vectors, which underlines the universality of this mathematical procedure

Key concepts

Wave Functions

In quantum mechanics, wave functions are essential to describe the quantum state of a particle or a system. The wave function, typically represented by \( \psi(x) \), provides a complete description of probabilities associated with the position and other properties of a particle.
- **Probabilistic Nature**: The absolute square of a wave function, \(|\psi(x)|^2\), represents the probability density of finding a particle at a specific position. Therefore, it is crucial that wave functions are properly normalized so that the total probability sums to one.
- **Types of Wave Functions**: In the context of this exercise, symmetric and antisymmetric wave functions play a vital role.
- **Symmetric Wave Functions**: These wave functions remain unchanged when particles are swapped. They are of the form \( \psi_{n}(x_1) \psi_{n}(x_2) + \psi_{n}(x_2) \psi_{n}(x_1) \).
- **Antisymmetric Wave Functions**: These wave functions change sign when the positions of particles are swapped. Such functions are expressed as \( \psi_{n}(x_1) \psi_{n}(x_2) - \psi_{n}(x_2) \psi_{n}(x_1) \). Antisymmetric wave functions are particularly significant for describing fermions, such as electrons.

Normalization

Normalization is an important process in quantum mechanics that ensures a wave function correctly represents the physical state it describes.
- **Purpose of Normalization**: By normalizing a wave function, we make sure that probabilities calculated from the function are accurate and meaningful. This involves adjusting the wave function so that the integral of its square equals one: \( \int |\psi(x)|^2 \, dx = 1 \).
- **Normalization Constant**: For symmetric and antisymmetric wave functions, the normalization constant is found by solving integrals over all space. - For symmetric or antisymmetric wave functions described in the exercise, the normalizing constant is \( 1/\sqrt{2} \). This ensures the double integral of the squared function equals one.
- **Implications of Normalization**: Once normalized, the wave functions can be used to calculate probabilities, expectation values, and other physical quantities accurately.
The process of normalization is not unique to quantum wave functions. As seen with the vector in the exercise, vectors also require a similar normalization approach to become unit vectors.

Quantum States

Quantum states are a cornerstone concept in quantum mechanics, representing the state of a system that encompasses all its properties.
- **Defining Quantum States**: A quantum state provides a full description of a quantum system's properties. These states can be represented by wave functions.
- **Orthogonality and Orthonormality**: In many quantum systems, states are chosen to be orthogonal and normalized (orthonormal).
- **Orthogonal States**: Two states are orthogonal if the integral of their product across all space equals zero. This means they do not overlap in terms of probability distribution.
- **Orthonormal States**: These states are both orthogonal and normalized, meaning their overlap is zero and their individual probability integrals equal one.
- **Superposition and Entanglement**: Quantum states can exist as a superposition of multiple states, contributing to unique phenomena such as quantum entanglement and interference. Understanding quantum states and how they are represented is vital for dealing with complex quantum systems, allowing us to predict and analyze physical outcomes effectively.