0
The hamiltonian operator is the observable corresponding to the total energy of the system. As with all observables it is given by a hermitian or self adjoint operator. This is true whether the hamiltonian is limited to momentum or contains potential.
Wiki User
∙ 16y ago
Add your answer:
Earn +20 pts
Why Hamilton's equations are called canonical equations?
The word canonical means "by a general law, rule, principle or criterion". When the Hamiltonian operator is applied to the (average momentum) wave function it gives quantized values. In this sense the Hamilton equations gives the Schrodinger equation discreet values by a general law.
What are Hamiltonian equations?
Hamiltonian equations are a representation of Hamiltonian mechanics. Please see the link.
What is Hamiltonian function?
The total energy of the system simply described in classical mechanics called as Hamiltonian.
Why is energy expressed as the second-order partial differential of a wave function in quantum mechanics?
You are referring to the Schrodinger Equation. This is because it comes from the classical view that the total energy is equal to the hamiltonian of a system:Kinetic Energy + Potential Energy = Total energy.Classically the kinetic energy is (1/2)mv2 = p2/(2m) ; where m is mass, v is velocity, p is momentum (p=mv).Now the momentum operator in QM is p=iħ∇ ;where ∇ is the gradient operator.This therefore yields the QM hamiltonian [-ħ2∇2/(2m) + V(x,y,z)]Ψ = EΨNow a more fun question to ask would be "Why is the Hamiltonian a function of the second-order partial differential with respect to position but the time dependent is only a function of a first-order differential with respect to time?"meaningHΨ = -iħ(dΨ/dt) or[-ħ2∇2/(2m) + V(x,y,z)]Ψ = -iħ(dΨ/dt)hint: Think Maxwell's Equations!
What is harmitian matrix?
Hermitian matrix (please note spelling): a square matrix with complex elements that is equal to its conjugate transpose.
What is the meaning of Hc in an Hamiltonian?
In mathematics and physics, H.c. means Hermitian conjugate. In a Hamiltonian in general, it is used to indicate that besides the previous terms you also have the Hermitian conjugate of those.
What is a Hermitian operator?
A Hermitian operator is any linear operator for which the following equality property holds: integral from minus infinity to infinity of (f(x)* A^g(x))dx=integral from minus infinity to infinity of (g(x)A*^f(x)*)dx, where A^ is the hermitian operator, * denotes the complex conjugate, and f(x) and g(x) are functions. The eigenvalues of hermitian operators are real and their eigenfunctions are orthonormal.
Expectation value of hermitian operator is real?
Yes. The derivation is a bit hard to do on here, but it can be found in Physical Chemistry: A Molecular Approach by McQuarrie and Simon.
Why Hamilton's equations are called canonical equations?
The word canonical means "by a general law, rule, principle or criterion". When the Hamiltonian operator is applied to the (average momentum) wave function it gives quantized values. In this sense the Hamilton equations gives the Schrodinger equation discreet values by a general law.
What is the definition of a Hermitian matrix?
Hermitian matrix defined:If a square matrix, A, is equal to its conjugate transpose, A†, then A is a Hermitian matrix.Notes:1. The main diagonal elements of a Hermitian matrix must be real.2. The cross elements of a Hermitian matrix are complex numbers having equal real part values, and equal-in-magnitude-but-opposite-in-sign imaginary parts.
What are Hamiltonian equations?
Hamiltonian equations are a representation of Hamiltonian mechanics. Please see the link.
What is the dimension of hermitian matrix?
77
What is the definition of a skew-Hermitian matrix?
Skew-Hermitian matrix defined:If the conjugate transpose, A†, of a square matrix, A, is equal to its negative, -A, then A is a skew-Hermitian matrix.Notes:1. The main diagonal elements of a skew-Hermitian matrix must be purely imaginary, including zero.2. The cross elements of a skew-Hermitian matrix are complex numbers having equal imaginary part values, and equal-in-magnitude-but-opposite-in-sign real parts.
What is the proper adjective for Hamilton?
Hamiltonian is the proper adjective for Hamilton. For instance: The Hamiltonian view on the structure of government was much different from that of Jefferson.
Is every unitary matrix hermitian?
Absolutely not. They are rather quite different: hermitian matrices usually change the norm of vector while unitary ones do not (you can convince yourself by taking the spectral decomposition: eigenvalues of unitary operators are phase factors while an hermitian matrix has real numbers as eigenvalues so they modify the norm of vectors). So unitary matrices are good "maps" whiule hermitian ones are not. If you think about it a little bit you will be able to demonstrate the following: for every Hilbert space except C^2 a unitary matrix cannot be hermitian and vice versa. For the particular case H=C^2 this is not true (e.g. Pauli matrices are hermitian and unitary).
What is Hamiltonian function?
The total energy of the system simply described in classical mechanics called as Hamiltonian.
How can you prove that ladder operators are hermitian operators?
You can't because they're not.
