Momentum Expectation Value Approaching 0

Oct 13, 2025 by windailys.com 15 views

Introduction

In the realm of quantum mechanics, the expectation value of momentum is a fundamental concept that plays a crucial role in understanding the behavior of particles. The expectation value of momentum is calculated using the momentum operator, which is given by the expression:

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

This expression is a mathematical representation of the momentum expectation value, where $\psi$ is the wavefunction of the particle, $\hbar$ is the reduced Planck constant, and $x$ is the position of the particle. In this article, we will explore the concept of momentum expectation value and its behavior when the wavefunction is real-valued.

Real-Valued Position Wavefunctions

When the wavefunction is real-valued, it means that the wavefunction does not have any imaginary components. In other words, the wavefunction can be represented as:

\psi(x) = \psi_R(x) + i\psi_I(x) = \psi_R(x)\tag{2}.

where $\psi_R(x)$ is the real part of the wavefunction and $\psi_I(x)$ is the imaginary part of the wavefunction. Since the wavefunction is real-valued, the expectation value of momentum can be simplified as:

\langle p\rangle = \int \psi^*\left(−i\hbar\frac{\partial}{\partial x}\right)\psi\, \mathrm dx = \int \psi_R\left(−i\hbar\frac{\partial}{\partial x}\right)\psi_R\, \mathrm dx\tag{3}.

Momentum Expectation Value Approaching 0

Now, let's consider the case where the momentum expectation value approaches 0. This means that the integral in equation (3) approaches 0 as the wavefunction approaches a constant value. Mathematically, this can be represented as:

\lim_{\psi \to \text{constant}} \langle p\rangle = \lim_{\psi \to \text{constant}} \int \psi_R\left(−i\hbar\frac{\partial}{\partial x}\right)\psi_R\, \mathrm dx = 0\tag{4}.

However, this result seems counterintuitive, as the momentum expectation value should not approach 0 for a real-valued wavefunction. This is because the momentum operator is a differential operator, and the derivative of a constant is 0. Therefore, one would expect the momentum expectation value to be 0 for a constant wavefunction.

Complex Numbers and the Momentum Operator

To resolve this apparent paradox, we need to consider the role of complex numbers in the momentum operator. The momentum operator is given by the expression:

\left(−i\hbar\frac{\partial}{\partial x}\right)\tag{5}.

This operator is a complex-valued operator, which means that it can produce complex-valued results. In particular, when the wavefunction is real-valued, the momentum operator will produce a complex-valued result.

The Role of the Imaginary Unit

The imaginary unit, denoted by $i$ , plays a crucial role in the momentum operator. The imaginary unit is defined as:

i = \sqrt{-1}\tag{6}.

This unit is used to represent complex numbers, which are essential in quantum mechanics. In particular, the momentum operator uses the imaginary unit to produce complex-valued results.

The Momentum Expectation Value for Real-Valued Wavefunctions

Now, let's consider the momentum expectation value for real-valued wavefunctions. Using the expression for the momentum operator, we can write:

\langle p\rangle = \int \psi^*\left(−i\hbar\frac{\partial}{\partial x}\right)\psi\, \mathrm dx = \int \psi_R\left(−i\hbar\frac{\partial}{\partial x}\right)\psi_R\, \mathrm dx\tag{7}.

Using the definition of the imaginary unit, we can rewrite this expression as:

\langle p\rangle = \int \psi_R\left(\hbar\frac{\partial}{\partial x}\right)\psi_R\, \mathrm dx\tag{8}.

This expression shows that the momentum expectation value for real-valued wavefunctions is proportional to the derivative of the wavefunction.

Conclusion

In conclusion, the momentum expectation value for real-valued wavefunctions is not necessarily 0. Instead, it is proportional to the derivative of the wavefunction. This result is consistent with the definition of the momentum operator, which uses the imaginary unit to produce complex-valued results. The role of complex numbers in the momentum operator is essential in resolving the apparent paradox of the momentum expectation value approaching 0 for real-valued wavefunctions.

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. North-Holland Publishing Company.

Q: What is the momentum expectation value?

A: The momentum expectation value is a measure of the average momentum of a particle in a quantum system. It is calculated using the momentum operator and the wavefunction of the particle.

Q: What is the momentum operator?

A: The momentum operator is a mathematical operator that represents the momentum of a particle. It is given by the expression:

\left(−i\hbar\frac{\partial}{\partial x}\right)\tag{1}.

Q: Why is the momentum operator complex-valued?

A: The momentum operator is complex-valued because it contains the imaginary unit, denoted by $i$ . The imaginary unit is used to represent complex numbers, which are essential in quantum mechanics.

Q: What happens to the momentum expectation value when the wavefunction is real-valued?

A: When the wavefunction is real-valued, the momentum expectation value is proportional to the derivative of the wavefunction. This is because the momentum operator uses the imaginary unit to produce complex-valued results.

Q: Why does the momentum expectation value not approach 0 for real-valued wavefunctions?

A: The momentum expectation value does not approach 0 for real-valued wavefunctions because the momentum operator is a differential operator. The derivative of a constant is 0, but the derivative of a non-constant function is not necessarily 0.

Q: What is the role of complex numbers in the momentum operator?

A: Complex numbers play a crucial role in the momentum operator. The imaginary unit, denoted by $i$ , is used to represent complex numbers, which are essential in quantum mechanics.

Q: Can you provide an example of a real-valued wavefunction?

A: Yes, a simple example of a real-valued wavefunction is the Gaussian wavefunction:

\psi(x) = \exp\left(-\frac{x^2}{2\sigma^2}\right)\tag{2}.

Q: How do you calculate the momentum expectation value for a real-valued wavefunction?

A: To calculate the momentum expectation value for a real-valued wavefunction, you need to use the expression:

\langle p\rangle = \int \psi^*\left(−i\hbar\frac{\partial}{\partial x}\right)\psi\, \mathrm dx = \int \psi_R\left(−i\hbar\frac{\partial}{\partial x}\right)\psi_R\, \mathrm dx\tag{3}.

Q: What is the significance of the momentum expectation value in quantum mechanics?

A: The momentum expectation value is a fundamental concept in quantum mechanics. It provides a measure of the average momentum of a particle in a quantum system, which is essential in understanding the behavior of particles.

Q: Can you provide a reference for further reading on this topic?

A: Yes, a good reference for further reading on this topic is the book "Quantum Mechanics" by Lev Landau and Evgeny Lifshitz.

Q: What is the relationship between the momentum expectation value and the wavefunction?

A: The momentum expectation value is related to the wavefunction through the momentum operator. The momentum operator acts on the wavefunction to produce the momentum expectation value.

Q: Can you provide a mathematical expression for the momentum expectation value?

A: Yes, the momentum expectation value can be expressed mathematically as:

\langle p\rangle = \int \psi^*\left(−i\hbar\frac{\partial}{\partial x}\right)\psi\, \mathrm dx\tag{4}.

Q: What is the significance of the imaginary unit in the momentum operator?

A: The imaginary unit, denoted by $i$ , plays a crucial role in the momentum operator. It is used to represent complex numbers, which are essential in quantum mechanics.

Q: Can you provide an example of a complex-valued wavefunction?

A: Yes, a simple example of a complex-valued wavefunction is the plane wave:

\psi(x) = \exp\left(ikx\right)\tag{5}.

Q: How do you calculate the momentum expectation value for a complex-valued wavefunction?

A: To calculate the momentum expectation value for a complex-valued wavefunction, you need to use the expression:

\langle p\rangle = \int \psi^*\left(−i\hbar\frac{\partial}{\partial x}\right)\psi\, \mathrm dx\tag{6}.

Q: What is the relationship between the momentum expectation value and the energy of a particle?

A: The momentum expectation value is related to the energy of a particle through the energy-momentum relation:

E^2 = p^2c^2 + m^2c^4\tag{7}.

Q: Can you provide a reference for further reading on the energy-momentum relation?

A: Yes, a good reference for further reading on the energy-momentum relation is the book "The Feynman Lectures on Physics" by Richard Feynman.