The Momentum Expectation Value

Oct 13, 2025 by windailys.com 97 views

Introduction

In the realm of quantum mechanics, the expectation value of momentum is a crucial concept that helps us understand the behavior of particles at the subatomic level. The expectation value of momentum is calculated using the wavefunction of the particle, which encodes all the information about the particle's state. However, when dealing with real-valued position wavefunctions, a peculiar phenomenon arises, leading to confusion in the momentum expectation value approaching 0. In this article, we will delve into the intricacies of this phenomenon and explore the underlying reasons behind it.

The Momentum Expectation Value

The momentum expectation value in a one-dimensional system is given by the following expression:

\langle p\rangle = \int \psi^*\left(-i\hbar\frac{\partial}{\partial x}\right)\psi\, \mathrm dx\tag{1}.

Here, $\psi$ is the wavefunction of the particle, $\psi^*$ is its complex conjugate, $i$ is the imaginary unit, $\hbar$ is the reduced Planck constant, and $x$ is the position coordinate. The momentum operator is represented by the differential operator $-i\hbar\frac{\partial}{\partial x}$ .

Real-Valued Position Wavefunctions

When the position wavefunction $\psi$ is real, the complex conjugate $\psi^*$ is equal to $\psi$ itself. Substituting this into the expression for the momentum expectation value, we get:

\langle p\rangle = \int \psi\left(-i\hbar\frac{\partial}{\partial x}\right)\psi\, \mathrm dx\tag{2}.

However, when we try to evaluate this integral, we encounter a problem. The momentum operator $-i\hbar\frac{\partial}{\partial x}$ is a differential operator that acts on the wavefunction $\psi$ . When $\psi$ is real, the product $\psi\left(-i\hbar\frac{\partial}{\partial x}\right)\psi$ is not necessarily real. In fact, it can be complex.

The Problem with Complex Numbers

The issue here is that complex numbers are not commutative. In other words, the order in which we multiply complex numbers matters. When we multiply a complex number by a differential operator, the result is not necessarily the same as when we multiply the differential operator by the complex number.

To see why this is a problem, let's consider a simple example. Suppose we have a real-valued wavefunction $\psi(x) = e^{-x^2}$ . When we apply the momentum operator to this wavefunction, we get:

\left(-i\hbar\frac{\partial}{\partial x}\right)\psi(x) = -2i\hbar xe^{-x^2}.

Now, when we multiply this result by the wavefunction $\psi(x)$ , we get:

\psi(x)\left(-i\hbar\frac{\partial}{\partial x}\right)\psi(x) = -2i\hbar xe^{-2x^2}.

As we can see, the result is complex. This is because the momentum operator $-i\hbar\frac{\partial}{\partial x}$ does not commute with the wavefunction $\psi(x)$ .

The Consequence of Non-Commutativity

The non-commutativity of complex numbers has a profound consequence for the momentum expectation value. When we try to evaluate the integral in equation (2), we are essentially trying to integrate a complex-valued function. However, the integral of a complex-valued function is not necessarily real.

In fact, the integral of a complex-valued function can be complex, and the result can depend on the specific form of the wavefunction. This means that the momentum expectation value can be complex, even when the wavefunction is real.

The Limit of Zero Momentum

Now, let's consider the limit of zero momentum. In this limit, the momentum expectation value approaches zero. However, when we try to evaluate the integral in equation (2), we encounter a problem. The integral is not necessarily real, and the result can depend on the specific form of the wavefunction.

In fact, the integral can be complex, and the result can depend on the specific form of the wavefunction. This means that the momentum expectation value can be complex, even in the limit of zero momentum.

Conclusion

References

[1] Dirac, P. A. M. (1958). The Principles of Quantum Mechanics. Oxford University Press.
[2] Landau, L. D., & Lifshitz, E. M. (1977). Quantum Mechanics: Non-Relativistic Theory. Pergamon Press.
[3] Messiah, A. (1961). Quantum Mechanics. John Wiley & Sons.

Q: What is the momentum expectation value, and why is it important in quantum mechanics?

A: The momentum expectation value is a measure of the average momentum of a particle in a quantum system. It is an important concept in quantum mechanics because it helps us understand the behavior of particles at the subatomic level. The momentum expectation value is calculated using the wavefunction of the particle, which encodes all the information about the particle's state.

Q: What is the problem with real-valued position wavefunctions and the momentum expectation value?

A: When the position wavefunction is real, the complex conjugate of the wavefunction is equal to the wavefunction itself. However, when we try to evaluate the integral for the momentum expectation value, we encounter a problem. The momentum operator does not commute with the wavefunction, and the result is a complex-valued function. This means that the momentum expectation value can be complex, even when the wavefunction is real.

Q: Why does the non-commutativity of complex numbers cause a problem for the momentum expectation value?

A: The non-commutativity of complex numbers means that the order in which we multiply complex numbers matters. When we multiply a complex number by a differential operator, the result is not necessarily the same as when we multiply the differential operator by the complex number. This causes a problem for the momentum expectation value because the momentum operator does not commute with the wavefunction.

Q: What is the consequence of the non-commutativity of complex numbers for the momentum expectation value?

A: The non-commutativity of complex numbers means that the momentum expectation value can be complex, even when the wavefunction is real. This means that the integral for the momentum expectation value is not necessarily real, and the result can depend on the specific form of the wavefunction.

Q: What happens to the momentum expectation value in the limit of zero momentum?

A: In the limit of zero momentum, the momentum expectation value approaches zero. However, when we try to evaluate the integral for the momentum expectation value, we encounter a problem. The integral is not necessarily real, and the result can depend on the specific form of the wavefunction. This means that the momentum expectation value can be complex, even in the limit of zero momentum.

Q: How can we resolve the confusion in the momentum expectation value approaching 0 for real-valued position wavefunctions?

A: To resolve the confusion in the momentum expectation value approaching 0 for real-valued position wavefunctions, we need to understand the underlying mathematics of the problem. We need to recognize that the non-commutativity of complex numbers causes a problem for the momentum expectation value, and we need to take this into account when evaluating the integral.

Q: What are some common misconceptions about the momentum expectation value approaching 0 for real-valued position wavefunctions?

A: Some common misconceptions about the momentum expectation value approaching 0 for real-valued position wavefunctions include:

Assuming that the momentum expectation value is always real
Assuming that the momentum operator commutes with the wavefunction
Assuming that the integral for the momentum expectation value is always real

Q: What are some real-world applications of the momentum expectation value approaching 0 for real-valued position wavefunctions?

A: The momentum expectation value approaching 0 for real-valued position wavefunctions has several real-world applications, including:

Understanding the behavior of particles in quantum systems
Developing new materials and technologies
Improving our understanding of the fundamental laws of physics

Q: What are some future directions for research on the momentum expectation value approaching 0 for real-valued position wavefunctions?

A: Some future directions for research on the momentum expectation value approaching 0 for real-valued position wavefunctions include:

Developing new mathematical tools to understand the behavior of complex-valued functions
Investigating the properties of complex-valued wavefunctions
Exploring the applications of complex-valued wavefunctions in quantum mechanics and beyond.

Conclusion

In conclusion, the momentum expectation value approaching 0 for real-valued position wavefunctions is a complex phenomenon that arises from the non-commutativity of complex numbers. The momentum operator does not commute with the wavefunction, and the result is a complex-valued function. This means that the momentum expectation value can be complex, even in the limit of zero momentum. By understanding the underlying mathematics of the problem, we can resolve the confusion in the momentum expectation value approaching 0 for real-valued position wavefunctions.