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Hermitian Operator & Particle in a Box | Physical Chemistry PDF Download

HERMITIAN OPERATOR

According to second postulate to every observable in classical mechanics, there corresponds an Hermitian operator in quantum mechanics.

A more general definition of Hermitian operator is:

If wave function are same,Hermitian Operator & Particle in a Box | Physical Chemistry

If any operator A satisfy above condition is a Hermitian operator.

Eigen values of Hermitian operator are real and eigen function of Hermitian operator are orthogonal.

 

Hermitian Operators:

Quantum mechanical operators are Hermitian operators. AS Hermitian operator  obeys the relation

Hermitian Operator & Particle in a Box | Physical Chemistry        Hermitian Operator & Particle in a Box | Physical ChemistryProperty of Hermitian operator

Where dq is volume element. For example ‘τ’ represents the Cartesian coordinates of a particle that can move in three dimensions, the integral is a three-fold integral and dq stands for (dxdydz).

The symbol Hermitian Operator & Particle in a Box | Physical Chemistry f* denotes the complex conjugate of the function f, and Hermitian Operator & Particle in a Box | Physical Chemistry A* denotes the complex conjugate of the operator A.  Complex quantities are surveyed briefly in Appendix B. If z is a complex quantity it can be written .

Hermitian Operator & Particle in a Box | Physical Chemistryz = x + iy

Where the real quantity ‘x’ is called the real part of ‘z’ and the real quantity of ‘y’ is called the imaginary part of ‘z’ . The complex conjugate of any complex number. Function, or operator is obtained by changing the sign of its imaginary part:

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

A real quantity or a real operator is equal to its complex conjugate, and imaginary quantity or an imaginary operator is equal to the negative of its complex conjugate.

Thus. We can write definition of Hermitian operator as an operator that satisfies the relation

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry ...........(1)

Where Hermitian Operator & Particle in a Box | Physical Chemistry is any well-behaved function.

Problem: Show that momentum operator is Hermitian

Sol. Momentum operator p = –ihd/dx

Substitution into equation (1)

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

And          Hermitian Operator & Particle in a Box | Physical Chemistry

Thus, we see that Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry  does, indeed, satisfy the equation (1). Therefore, the momentum operator is a Hermitian operator.

If follows from the definition of Hermitian operators that an operator  is Hermitian. When it is its own Hermitian adjoint – that is Hermitian Operator & Particle in a Box | Physical Chemistry . On the other hand an operator  is its complex conjugate – that is Hermitian Operator & Particle in a Box | Physical Chemistry at = at

Here are some relationships concerning Hermitian adjoints:

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry

  

Problem: Show that Px is Hermitian by using concept of Hermitian adjoint.

Soln. AS we know any operator is hermitian only if    

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry      Hermitian Operator & Particle in a Box | Physical Chemistry

AVERAGE VALUE OR EXPECTATIONN VALUE

If function Hermitian Operator & Particle in a Box | Physical Chemistry ψ is normalized than Hermitian Operator & Particle in a Box | Physical Chemistry Hermitian Operator & Particle in a Box | Physical Chemistry

SCHRODINGER EQUATION

The operator corresponding to the energy E is called the Hamiltonian operator and is represented by the symbol HOP. Thus we have.

Hermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry......(5)
Hermitian Operator & Particle in a Box | Physical Chemistry      ...................(6)

Now according to equation (2) we have.

Hermitian Operator & Particle in a Box | Physical Chemistry       .................(7)

Where E is the eigenvalue of the Hamiltonian operator HOP a quantity which has a precise value for a given state. Thus, Hermitian Operator & Particle in a Box | Physical Chemistry

Hermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry    ....................(8)

It is called schrodinger wave equation.


APPLICATION OF SCHRODINGER EQUATION:

Particle in a box → Translational energy → Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

Simple Harmonic Oscillation → Vibrational energy → Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

Rigid rotor → Rotational energy → Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

Hydrogen atom → Electronic energy → Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry Potential Energy

  

PARTICLE IN A ONE DIMENSIONAL BOX

In this problem a particle of mass is placed in a one dimensional box of length/particle is free to move. Box have infinite high walls.

It is assumed that the potential energy of particle is zero every where inside the box.

That is potential          V(x)=0

Thus the one dimensional schrodinger equation, we have.

Hermitian Operator & Particle in a Box | Physical Chemistry

Neglecting E is comparison to ∞, we have.

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

That is Hermitian Operator & Particle in a Box | Physical Chemistry outside the box. This means that the particle cannot exist outside the region 0 < x < 1. Within the box, the schrodinger equation for the motion of the particle takes the form.

Hermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry                   .................(2)
Where, Hermitian Operator & Particle in a Box | Physical Chemistry  .................(3)

A general solution of equation (2) is given by.

 Hermitian Operator & Particle in a Box | Physical Chemistry.................(2)

Where A and B are constant.

Only those function which satisfy the boundary conditions if Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry will be acceptable wave functions for ψ = 0 at x = 0 Hermitian Operator & Particle in a Box | Physical Chemistry equation (4) becomes.

Hermitian Operator & Particle in a Box | Physical Chemistry

The above expression will be true only where αl is an integral multiple of π that is,

Hermitian Operator & Particle in a Box | Physical Chemistry  ...........(6)

When n can have only integral values of 1,2,3,……….A value of n=0 is eliminated since it leads to α=0 or Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistryevery where within the box. Substituting α from equation (6) in equation (5) we get.

Hermitian Operator & Particle in a Box | Physical Chemistry..................(7)
Now from equation (3) we have.

Hermitian Operator & Particle in a Box | Physical Chemistry......................(8)

Substituting the value of α from equation (6) we have, Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

Hermitian Operator & Particle in a Box | Physical Chemistry.........(9)

In this case, n represents the quantum number.

So energy difference between the two successive energy levels.

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

As we knowHermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry
so Hermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry

 

ZERO POINT ENERGY (Ground state energy):

For ground state energy we will put n = 1 in the equation.

Hermitian Operator & Particle in a Box | Physical Chemistry      n = 1Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

 

PARTICLE IN TWO DIMENSIONAL BOX

For 2-D box wave function Hermitian Operator & Particle in a Box | Physical Chemistryψ  will dependent upon two independent variable x and y and Hermitian Operator & Particle in a Box | Physical Chemistry is the multiplication of both function. Hermitian Operator & Particle in a Box | Physical Chemistry

Schrodinger wave equation for free particle for 2-D box:

Hermitian Operator & Particle in a Box | Physical Chemistry

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

Dividing both side by X(x), Y(y).

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry
 

The term Hermitian Operator & Particle in a Box | Physical Chemistry and Hermitian Operator & Particle in a Box | Physical Chemistry in the above equation is a constant quality. L.H.S must also be a constant quantity and x, y both are independent on the other term and each is equal to a constant quantity.

Hermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry......................(1)
Hermitian Operator & Particle in a Box | Physical Chemistry......................(2)
 

Hermitian Operator & Particle in a Box | Physical ChemistrySolution of equation (1), Hermitian Operator & Particle in a Box | Physical Chemistry  Hermitian Operator & Particle in a Box | Physical Chemistry

Solution of equation (2), Hermitian Operator & Particle in a Box | Physical Chemistry Hermitian Operator & Particle in a Box | Physical Chemistry

DEGENERACY

More than one state with equal energy
Hermitian Operator & Particle in a Box | Physical Chemistry

Hermitian Operator & Particle in a Box | Physical Chemistry

Hermitian Operator & Particle in a Box | Physical Chemistry   Hermitian Operator & Particle in a Box | Physical Chemistry

Zero Point Energy:
nx = 1 ny = 1           Hermitian Operator & Particle in a Box | Physical Chemistry
nx = 1 ny = 2        Hermitian Operator & Particle in a Box | Physical Chemistry

PARTICLE IN THREE DIMENSIONAL BOX

Now we will discuss the motion of particle of mass m in three dimensional box. As in one dimensional box in three dimensional box also, the potential energy is zero with in the box and infinite outside the box.

So three dimensional schrodinger equation.

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

Where the function Hermitian Operator & Particle in a Box | Physical Chemistryψ  will depend upon three independent variable x, y, z to solve the above equation we write the function Hermitian Operator & Particle in a Box | Physical Chemistry ψ  as the product of three wave function.

Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry

Where, Hermitian Operator & Particle in a Box | Physical ChemistryX(x), Y(y), Z(z) are three function.

Put this value in equation (1).

Hermitian Operator & Particle in a Box | Physical Chemistry.......(2)

Hermitian Operator & Particle in a Box | Physical Chemistry .......(3)

Dividing by XYZ, we have.

Hermitian Operator & Particle in a Box | Physical Chemistry.......(4)

The term Hermitian Operator & Particle in a Box | Physical Chemistryα2 in the above equation is a constant quantity. Hence the sum of the three terms on the left hand side of equation (4) must also be a constant quantity. If we change the value of x (or y or z) keeping the other two variables constants even then the above constancy has to be satisfied. This is possible only when each term is independent of the other term and each is equal to a constant quantity so that the sum of three constant is equal to Hermitian Operator & Particle in a Box | Physical Chemistry .α2

So we write.

Hermitian Operator & Particle in a Box | Physical Chemistry.......(5)
Hermitian Operator & Particle in a Box | Physical Chemistry.......(6)
Hermitian Operator & Particle in a Box | Physical Chemistry .......(7)

Where,         Hermitian Operator & Particle in a Box | Physical Chemistry     Hermitian Operator & Particle in a Box | Physical Chemistry    .......(8)
Hermitian Operator & Particle in a Box | Physical Chemistry         .......(9)
Hermitian Operator & Particle in a Box | Physical Chemistry        .......(10)                  Hermitian Operator & Particle in a Box | Physical Chemistry

With,               Hermitian Operator & Particle in a Box | Physical ChemistryHermitian Operator & Particle in a Box | Physical Chemistry .......(11) 

And               E = Ex + Ey + Ez  Hermitian Operator & Particle in a Box | Physical Chemistry .......(12) 

Now we have three separate equations to be solved each of them has a form of one-dimensional box. Thus the normalized wave function of a three-dimensional box is

Hermitian Operator & Particle in a Box | Physical Chemistry
Hermitian Operator & Particle in a Box | Physical Chemistry..........(13)

The constant Hermitian Operator & Particle in a Box | Physical Chemistry will be given by.

Hermitian Operator & Particle in a Box | Physical Chemistry..........(14)

And total energy s Hermitian Operator & Particle in a Box | Physical Chemistry       E = Ex + Ey + Ez  Hermitian Operator & Particle in a Box | Physical Chemistry= Hermitian Operator & Particle in a Box | Physical Chemistry        ..........(15)

There are three quantum numbers one each for energy every degree of freedom.

The document Hermitian Operator & Particle in a Box | Physical Chemistry is a part of the Chemistry Course Physical Chemistry.
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FAQs on Hermitian Operator & Particle in a Box - Physical Chemistry

1. What is a Hermitian operator in quantum mechanics?
Ans. In quantum mechanics, a Hermitian operator is an operator that satisfies the condition of being self-adjoint. It means that the operator is equal to its own adjoint (conjugate transpose). In other words, if A is a Hermitian operator, then A† = A, where A† represents the adjoint of A. Hermitian operators play a crucial role in quantum mechanics as they have real eigenvalues and their eigenvectors form a complete orthonormal basis.
2. How is the Hermitian operator related to the particle in a box problem?
Ans. The particle in a box is a fundamental problem in quantum mechanics that involves studying a particle confined to a one-dimensional box. In this problem, the Hamiltonian operator, which represents the total energy of the system, is a Hermitian operator. The eigenvalues and eigenvectors of the Hamiltonian operator provide information about the energy levels and wave functions of the particle in the box. Thus, the Hermitian nature of the Hamiltonian operator ensures the physical significance and observability of the energy levels in the particle in a box problem.
3. What are the properties of Hermitian operators?
Ans. Hermitian operators possess several important properties in quantum mechanics. Some of these properties include: - Hermiticity: The operator is equal to its adjoint (conjugate transpose), i.e., A† = A. - Real eigenvalues: The eigenvalues of a Hermitian operator are always real numbers. - Orthogonal eigenvectors: The eigenvectors corresponding to distinct eigenvalues of a Hermitian operator are orthogonal to each other. - Completeness: The set of eigenvectors of a Hermitian operator forms a complete orthonormal basis for the underlying Hilbert space.
4. How can Hermitian operators be measured in experiments?
Ans. In experiments, the measurement of a Hermitian operator is carried out by determining the corresponding eigenvalues of the operator. This can be done by preparing a system in a superposition of the eigenstates of the Hermitian operator and then measuring the observable associated with the operator. The measured result will correspond to one of the eigenvalues, and the probability of obtaining a specific eigenvalue is given by the squared magnitude of the projection of the system's state onto the corresponding eigenvector.
5. Can all operators in quantum mechanics be Hermitian?
Ans. No, not all operators in quantum mechanics are Hermitian. While Hermitian operators have important physical properties, such as real eigenvalues, there are operators that do not satisfy the condition of being self-adjoint. For example, non-Hermitian operators can have complex eigenvalues, which are associated with systems that exhibit non-unitary time evolution or energy dissipation. Non-Hermitian operators have also been studied extensively in various areas of physics, including quantum mechanics beyond standard Hermitian formulations.
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