The Hamiltonian Operator

The Hamiltonian operator is a quantum mechanical operator with energy as eigenvalues. It corresponds to the total energy inside a system including kinetic and potential energy. The eigenvalues of this operator are, in fact, the possible outcomes of the total energy of a quantum mechanical system. Being able to calculate, a very difficult parameter, energy, the Hamiltonian operator is a fundamental operator in quantum mechanical and theoretical equations.

According to the second postulate of quantum mechanics;

  • For every measurable property of a quantum mechanical system such as position, momentum, energy, etc, there exists a corresponding operator. For example, Linear operator, momentum operator, Hamiltonian operator (H), Laplacian operator (2), etc. 

A hamiltonian operator is the energy operator for wave function (ψ) in time-independent Schrodinger’s wave theory equation. It combines the operators for kinetic energy and potential energy to make a time-independent Hamiltonian operator formula equation.

Hamiltonian Operator - Quantum equation - Schrodinger equation formula

Operation on wave function (ψ) produces Schrodinger’s wave equation. Operating Hamiltonian operator (H) over the wave function produces fixed values that correspond to the eigenvalues of energy.

The wave function (ψ) is the wave representing a spatial distribution of particles. For example, electrons in an atom are described by a wave function centered on the nucleus. A wave function does not describe the exact location of the electron. It tells us the probability of finding the electron at a given point. These points are the eigenvalues of the Hamiltonian operator.

Hamiltonian: A Quantum Mechanical Operator

The Hamiltonian operator was introduced by Sir William Rowan Hamilton (1805 – 1865). The Hamiltonian operator, also known as the total energy operator is represented by Ĥ or simply H. This operator comes from his formulation of classical mechanics that is observed on the total energy for the particle in a box. This quantum mechanical operator provided the basic framework for the Hamiltonian mechanics which was the reformulation of the Langrangian mechanics.

The Hamiltonian operator is a hermitian operator as it fulfills all its conditions.

Ĥ = -ħ2/2m . ∂2/∂x2 + V(x)

where,

2/2m . /x = Kinetic energy operator

V(x) = Potential energy operator

In other words, the hamiltonian operator can be converted into different formulas depending on the functions it is being applied to.

Hamiltonian Operator for 3-D box

Hamiltonian operator for a three-dimensional box, where potential energy is zero, can be written as;

Ĥ = Ĥx + Ĥy + Ĥz

where,

Ĥx = -ħ2/2m . ∂2/∂x2        Ĥy = -ħ2/2m . ∂2/∂y2        Ĥ= -ħ2/2m . ∂2/∂z2

Hamiltonian operator for (n)-particles

The Hamiltonian operator for n number of particles can be formulated by summation of kinetic and potential energies of all n-particles.

Ĥ = Tn+Vn   ———-   eq (1)

where,

T = Kinetic energy operator symbol

V = Potential energy operator symbol

such that,

V(n)=V(r1+r2+….+rn,t)   ———-   eq (2)

and

T(n) = pn2/2mn = -ħ2/2mnn2   ———-   eq (3)

where,

 = Laplacian operator = /x

Now, putting eq (2) and eq (3) in eq (1), we get,

Ĥ = pn2/2mn + V(r1+r2+….rn,t)

Or,

Ĥ = -ħ2/2 . 1/mn . n2 + V (r1+r2+….rn,t)

This is a form of Hamiltonian operator for n number of particles.

Related topics

Application of Hamiltonian operator

The energy spectrum systems and time evolution complex equations are fundamentally dependent on the Hamiltonian operator. For example, time-independent Schrodinger’s wave equation.

Time-independent Schrodinger wave equation

According to the time-independent Schrodinger wave equation.

Ĥ Ψ  =  E Ψ

This equation states that the hamiltonian operator acts upon the wave function (Ψ) and we get back the same function with some constant values of (E), the energy. The wave function (Ψ) is, therefore, its eigenfunction, and energy (E) is its eigenvalue.

Explanation

Consider a particle confined in a box of fixed length i.e. [0 to a]. Inside this box, the potential energy V is taken to be zero, because the particle is in continuous motion, while outside the box, the potential energy is infinite. 

A brief derivation of the one-dimensional particle in a box model is given here to illustrate the use of the Hamiltonian operator.

Applying the Hamiltonian operator Schrodinger’s equation Changes to;

2/2m . 2 Ψ  = E Ψ   ———-   eq (1)

The solution of this differential equation is of the form:

Ψ = A.sin(kx) + B.cos(kx)   ———-   eq (2)

Two conditions arise here due to the boundary conditions.

  • First, Ψ = 0 at x = 0, so, B must equal to zero.
  • Second, Ψ = 0 at x = a, k = nπ/a where n is a positive integer.

When x = 0, Ψ = 0, the eq (2) becomes;

0 = 0 + B(1)

B = 0

When x = a, Ψ = 0, the eq (2) becomes; 

0 = Asinka

Since A≠0 so,

sin (ka) = 0

ka = sin-1(0)

ka = nπ

k = nπ/a

Putting values in eq (2), we get,

Ψ = Asin (nπx/a)

The constant A is determined to be 2/a by normalizing the wave function. To find the energy the general solution Ψ is substituted back to the Schrodinger equation.

 2/2m . 2 Ψ  =  E Ψ

2/2m . 2Asin(nπx/a) = E.Asin (nπx/a)

From the above equation, it can be shown that E = n2h2/8ma2. Since (n) is only to take allowed integer values, this shows that the energy levels of the system are quantized, thanks to the Hamiltonian operator.

Key Takeaway(s)

Hamiltonian operator exists in two forms:

  1. Time-independent Hamiltonian operator
  2. Time-dependent Hamiltonian operator

Hamiltonian Operator time dependent and independent

Concepts Berg

Explain the significance of Hamiltonian?

Hamiltonian is an operator for the total energy of a system in quantum mechanics. It tells about kinetic and potential energy for a particular system. The solution of Hamiltonians equation of motion will yield a trajectory in terms of position and momentum as a function of time. Moreover, it is a significant equation as it leads quantum mechanics to Hamiltonian mechanics, as an alternative to Langrangian mechanics.

How to check whether an operator is a Hermitian?

For the operator to be hermitian, it must fulfill the following conditions.

  1. It justifies the matrix representation as conjugate transpose. In other words, it can be flipped over to the other side.
  2. The eigenvalues of the hermitian operator are real (quantized) values. 
  3. The sign function of the hermitian operator with different eigenvalues is orthogonal (orthonormal).

Is Hamiltonian a hermitian operator?

For the operator to be hermitian, it must fulfill the following conditions.

  1. If Ψ1 and Ψ2 are two functions and A is an operator then,

Ψ1* ( Â.Ψ2 ) . dτ = Ψ2 ( Â.Ψ1* ) . dτ

  1. The eigenvalues of the hermitian operator are real values. 
  2. The sign function of the hermitian operator with different eigenvalues is orthogonal.

Since Hamiltonians obey all these conditions, it is a ‘hermitian operator’.

What’s so special about the Hamiltonian?

Hamiltonian is an energy operator and tells us about the total energy value of a quantum mechanical system.

How is the hamiltonian operator determined?

According to the time-independent Schrodinger wave equation, the Hamiltonian is the sum of kinetic energy and potential energy.

Ĥ = -ħ2/2m /x + V(x)

where,

2/2m /x = K.E operator and V(x) = P.E operator

Hamiltonian acts on given eigen functions i.e. wave function (Ψ) to give eigen values (E).

Ĥ Ψ  =  E Ψ

How do you use a Hamiltonian operator?

 Hamiltonian is the sum of kinetic energy and potential energy.

Ĥ=-ħ2/2m /x + V(x)

We can also use the Hamiltonian operator in the expectation value. 

<H> = ∫Ψn*ĤΨn.dx

As

Ĥ Ψ  =  E Ψ

<H> = ∫Ψn*EnΨndx

<H> = En∫Ψn*Ψndx

As ∫Ψn*Ψndx=1 so

<H>=E(n)

What is the hamiltonian operator formula?

It is the sum of kinetic energy and potential energy.

Ĥ = -ħ2/2m /x + V(x)

What is the hamiltonian operator symbol?

The total energy operator called the hamiltonian operator is represented by Ĥ, Ȟ or, simply by H.

What is the Hamiltonian in quantum mechanics?

In quantum mechanics, the Hamiltonian is an operator for the total energy of a system. It includes both the kinetic and potential energy of the system. 

Ĥ = -ħ2/2m /x + V(x)

What is the relationship between hamiltonian operators and eigenvectors?

When an operator acts upon a function and gets back the same function with some constant value, the function is called an eigenfunction or eigenvector, and the constant value is called an eigenvalue.

The relationship between the hamiltonian operator and eigenfunction is that when hamiltonian acts upon a function (Ψ), we will get back this function (eigenfunction) with some quantized energy value (eigenvalue).

What is the Hamiltonian operator for hydrogen?

When we use two masses instead of one mass in the equation, then ‘m’ is replaced by ‘μ’ which is the reduced mass.

Ĥ=-ħ2/2μ  /x + V(x)

where,

μ = (m1m2) / (m1+m2)

References

  • Physical Chemistry | A molecular approach by Donald A. McQuarrie (University of California, Davis) and John D. Simon (DUKE University)
  • Hamiltonian operators for molecules by C.-K. Skylaris (University of Southampton, School of Chemistry)
  • The Hamiltonian (Hyperphysics, Department of Physics and Astronomy, Georgia State University)
  • Operators in Quantum Mechanics (Hyperphysics, Department of Physics and Astronomy, Georgia State University)

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