Expectation Value Of Potential Energy For Hydrogen Atom

In quantum mechanics, the hydrogen atom serves as a fundamental system for understanding atomic structure and behavior. One critical aspect of analyzing quantum systems is determining the expectation value of various physical quantities, such as potential energy. The expectation value provides insight into the average value of a property that one would expect to obtain from a large number of measurements. For the hydrogen atom, calculating the expectation value of the potential energy reveals essential characteristics of its electronic structure.

Understanding the Hydrogen Atom

The hydrogen atom, composed of a single proton and a single electron, is the simplest atom in nature. Its simplicity allows for precise theoretical treatment using the Schrödinger equation, making it a cornerstone for understanding more complex atoms. The potential energy in the hydrogen atom arises from the electrostatic interaction between the positively charged proton and the negatively charged electron.

Defining Expectation Value

In quantum mechanics, the expectation value of an operator, such as potential energy, is defined as the average result of measurements on a system in a specific quantum state. Mathematically, for a system in a state described by the wavefunction ψ, the expectation value of an operator  is given by:

⟨Â⟩ = ∫ ψ^* Â ψ dτ

Where:

• ψ^* is the complex conjugate of the wavefunction ψ.
• Â is the operator corresponding to the physical quantity of interest.
• dτ is the volume element in the appropriate coordinate system.
• The integral is taken over all space.

Potential Energy Operator for the Hydrogen Atom

The potential energy V due to the electrostatic interaction between the proton and electron in the hydrogen atom is given by Coulomb's law:

V(r) = -e^2 / (4πε₀r)

Where:

• e is the elementary charge.
• ε₀ is the vacuum permittivity.
• r is the distance between the proton and the electron.

In atomic units, where e = 1, 4πε₀ = 1, the potential energy simplifies to:

V(r) = -1/r

Wavefunctions of the Hydrogen Atom

The wavefunctions for the hydrogen atom are solutions to the time-independent Schrödinger equation in spherical coordinates (r, θ, φ). These wavefunctions can be expressed as:

ψ(r, θ, φ) = R(n,l)(r) * Y(l,m)(θ, φ)

Where:

• R(n,l)(r) is the radial wavefunction, which depends on the principal quantum number n and the azimuthal quantum number l.
• Y(l,m)(θ, φ) is the spherical harmonic function, which depends on the azimuthal quantum number l and the magnetic quantum number m.

The principal quantum number n determines the energy level of the electron, with n = 1, 2, 3, ... corresponding to the ground state, first excited state, second excited state, and so on. The azimuthal quantum number l determines the shape of the electron's orbital and ranges from 0 to n-1. The magnetic quantum number m determines the orientation of the orbital in space and ranges from -l to +l.

Calculating the Expectation Value of Potential Energy

To calculate the expectation value of the potential energy, we need to evaluate the integral:

⟨V⟩ = ∫ ψ^* V(r) ψ dτ

In spherical coordinates, the volume element dτ is given by:

dτ = r^2 sin(θ) dr dθ dφ

Thus, the expectation value of the potential energy becomes:

⟨V⟩ = ∫₀^∞ ∫₀^π ∫₀^(2π) ψ^* (r, θ, φ) V(r) ψ(r, θ, φ) r^2 sin(θ) dr dθ dφ

Substituting the expression for the potential energy V(r) = -1/r, we get:

⟨V⟩ = -∫₀^∞ ∫₀^π ∫₀^(2π) ψ^* (r, θ, φ) (1/r) ψ(r, θ, φ) r^2 sin(θ) dr dθ dφ ⟨V⟩ = -∫₀^∞ ∫₀^π ∫₀^(2π) ψ^* (r, θ, φ) ψ(r, θ, φ) r sin(θ) dr dθ dφ

Let's consider the ground state of the hydrogen atom, where n = 1, l = 0, and m = 0. The wavefunction for the ground state is:

ψ₁₀₀(r, θ, φ) = (1/√(πa₀³)) e^(-r/a₀)

Where a₀ is the Bohr radius.

The expectation value of the potential energy for the ground state is:

⟨V⟩ = -∫₀^∞ ∫₀^π ∫₀^(2π) (1/√(πa₀³)) e^(-r/a₀) (1/√(πa₀³)) e^(-r/a₀) r sin(θ) dr dθ dφ ⟨V⟩ = -(1/πa₀³) ∫₀^∞ ∫₀^π ∫₀^(2π) e^(-2r/a₀) r sin(θ) dr dθ dφ

We can separate the integral into radial and angular parts:

⟨V⟩ = -(1/πa₀³) [∫₀^∞ r e^(-2r/a₀) dr] [∫₀^π sin(θ) dθ] [∫₀^(2π) dφ]

Evaluating the integrals: ∫₀^∞ r e^(-2r/a₀) dr = a₀²/4 ∫₀^π sin(θ) dθ = 2 ∫₀^(2π) dφ = 2π

Substituting these results back into the expression for ⟨V⟩: ⟨V⟩ = -(1/πa₀³) (a₀²/4) (2) (2π) ⟨V⟩ = -a₀²/a₀³ ⟨V⟩ = -1/a₀

In atomic units, a₀ = 1, so the expectation value of the potential energy for the ground state of the hydrogen atom is: ⟨V⟩ = -1

General Formula for Expectation Value of Potential Energy

For a hydrogen atom in the state |n, l, m⟩, the expectation value of the potential energy can be found using the general formula:

⟨V⟩ = -e^2 / (4πε₀) ⟨1/r⟩ = -Z e²/ (a₀ n²)

Where:

• Z is the atomic number (Z=1 for hydrogen).
• a₀ is the Bohr radius.
• n is the principal quantum number.

This formula shows that the expectation value of the potential energy is inversely proportional to the square of the principal quantum number, n. As n increases, the electron is, on average, farther from the nucleus, and the potential energy becomes less negative (closer to zero).

Detailed Step-by-Step Calculation

1. Define the potential energy operator: V(r) = -e²/ (4πε₀r)
2. Write the general form of the expectation value: ⟨V⟩ = ∫ ψ^* V(r) ψ dτ
3. Substitute the potential energy operator: ⟨V⟩ = ∫ ψ^* (-e²/ (4πε₀r)) ψ dτ
4. Use the hydrogen atom wavefunction ψ(r, θ, φ) = R(n,l)(r) * Y(l,m)(θ, φ): ⟨V⟩ = ∫₀^∞ ∫₀^π ∫₀^(2π) [R(n,l)(r) * Y(l,m)(θ, φ)]^* (-e²/ (4πε₀r)) [R(n,l)(r) * Y(l,m)(θ, φ)] r² sin(θ) dr dθ dφ
5. Separate the integral: ⟨V⟩ = (-e²/ (4πε₀)) ∫₀^∞ R(n,l)^(r) R(n,l)(r) (1/r) r² dr ∫₀^π ∫₀^(2π) Y(l,m)^(θ, φ) Y(l,m)(θ, φ) sin(θ) dθ dφ
6. Normalize the spherical harmonics: ∫₀^π ∫₀^(2π) Y(l,m)^*(θ, φ) Y(l,m)(θ, φ) sin(θ) dθ dφ = 1
7. Simplify the expression: ⟨V⟩ = (-e²/ (4πε₀)) ∫₀^∞ R(n,l)^*(r) R(n,l)(r) r dr
8. Evaluate the radial integral: This integral depends on the specific radial wavefunction R(n,l)(r). For example, for the ground state (n=1, l=0), R₁₀(r) = 2(Z/a₀)^(3/2) e^(-Zr/a₀).
9. Substitute the radial wavefunction and evaluate the integral for the ground state: ⟨V⟩ = (-e²/ (4πε₀)) ∫₀^∞ [2(Z/a₀)^(3/2) e^(-Zr/a₀)]^2 r dr ⟨V⟩ = (-e²/ (4πε₀)) 4(Z/a₀)³ ∫₀^∞ e^(-2Zr/a₀) r dr
10. Solve the integral: ∫₀^∞ r e^(-2Zr/a₀) dr = (a₀/2Z)²
11. Substitute the result back into the expression: ⟨V⟩ = (-e²/ (4πε₀)) 4(Z/a₀)³ (a₀/2Z)² ⟨V⟩ = (-e²/ (4πε₀)) 4 (Z³/a₀³) (a₀²/4Z²) ⟨V⟩ = (-e²/ (4πε₀)) (Z/a₀)
12. For hydrogen, Z=1: ⟨V⟩ = -e² / (4πε₀a₀)

Using the fact that the energy of the ground state E₁ = -e² / (8πε₀a₀), we have: ⟨V⟩ = 2E₁

This shows that the expectation value of the potential energy is twice the ground state energy of the hydrogen atom.

Physical Interpretation

The expectation value of the potential energy represents the average potential energy of the electron in the hydrogen atom. For the ground state, the negative value indicates that the electron is bound to the nucleus, and energy is required to remove it. As the principal quantum number n increases, the expectation value of the potential energy becomes less negative, indicating that the electron is, on average, farther from the nucleus and more weakly bound.

Virial Theorem

The virial theorem provides a useful relationship between the expectation values of the kinetic energy T and the potential energy V for a system in equilibrium. For the hydrogen atom, the virial theorem states:

2⟨T⟩ = -⟨V⟩

This theorem implies that the expectation value of the kinetic energy is half the magnitude of the expectation value of the potential energy. Since the total energy E is the sum of the kinetic and potential energies:

E = ⟨T⟩ + ⟨V⟩

Using the virial theorem, we can express the total energy in terms of the potential energy:

E = -⟨V⟩/2 + ⟨V⟩ = ⟨V⟩/2

This relationship further illustrates the importance of the potential energy in determining the overall energy of the hydrogen atom.

Importance of Understanding Expectation Values

Calculating and understanding expectation values is crucial for several reasons:

1. Predicting Measurement Outcomes: Expectation values provide predictions for the average results of measurements on quantum systems.
2. Understanding Atomic Structure: They offer insights into the electronic structure and behavior of atoms and molecules.
3. Validating Theoretical Models: Comparing theoretical predictions with experimental measurements helps validate and refine quantum mechanical models.
4. Applications in Quantum Chemistry and Physics: Expectation values are used extensively in quantum chemistry and physics to calculate various properties of atoms and molecules.

Example Calculation for n=2 State

For the n=2 state, we have four possible states: (2,0,0), (2,1,-1), (2,1,0), and (2,1,1). Let’s calculate the expectation value of the potential energy for the state (2,0,0).

The radial wavefunction for the (2,0) state is: R₂₀(r) = 1/√(2a₀³) (1 - r/(2a₀)) e^(-r/(2a₀))

The expectation value of V is: ⟨V⟩ = ∫₀^∞ R₂₀^*(r) (-1/r) R₂₀(r) r² dr ⟨V⟩ = - ∫₀^∞ (1/√(2a₀³) (1 - r/(2a₀)) e^(-r/(2a₀)))^2 r dr ⟨V⟩ = - 1/(2a₀³) ∫₀^∞ (1 - r/(2a₀))^2 e^(-2r/(2a₀)) r dr ⟨V⟩ = - 1/(2a₀³) ∫₀^∞ (1 - r/a₀ + r²/(4a₀²)) e^(-2r/(2a₀)) r dr

Solve the integral: ∫₀^∞ r e^(-2r/(2a₀)) dr = a₀²/4 ∫₀^∞ r² e^(-2r/(2a₀)) dr = a₀³/4 ∫₀^∞ r³ e^(-2r/(2a₀)) dr = a₀⁴/2

⟨V⟩ = - 1/(2a₀³) [a₀²/4 - a₀³/4a₀ + a₀⁴/(4a₀² * 2)] ⟨V⟩ = - 1/(2a₀³) [a₀²/4 - a₀²/4 + a₀²/8] ⟨V⟩ = - 1/(2a₀³) (a₀²/8) ⟨V⟩ = - 1/(16a₀)

Converting to atomic units (a₀ = 1): ⟨V⟩ = -1/4

So, for the n=2 state, the expectation value of the potential energy is -1/4 atomic units. This confirms the general formula ⟨V⟩ = -Z/n^2, where for n=2 and Z=1, ⟨V⟩ = -1/4.

Conclusion

The expectation value of the potential energy for the hydrogen atom provides essential insights into its electronic structure and behavior. By understanding how to calculate and interpret these expectation values, one can gain a deeper understanding of quantum mechanics and its applications in atomic and molecular physics. The step-by-step calculations and physical interpretations discussed in this article offer a comprehensive guide for students, researchers, and anyone interested in exploring the quantum world of the hydrogen atom.