Please help -Dirac delta potential-, Hermitian Conjugate

In summary, the conversation is about solving a problem from Griffiths' book on quantum mechanics involving the allowed energy for a double Dirac potential. The person is asking for help in verifying if they are on the right track with their solution. The conversation then delves into the properties of a Hermitian conjugate and how it relates to inner products. The question of whether or not <blT+la>* = <alT+lb> is always true, regardless of T being Hermitian, is raised.
  • #1
enalynned
7
0
Please help! -Dirac delta potential-, Hermitian Conjugate

Im trying to solve problem 2.26 from Griffiths (1st. ed, Intro to Q.M.). Its about the allowed energy to double dirac potential. I came up with a final equation that is trancedental. (After I separate the even and odd solution of psi.) Am I on the right track?

Please refer to Griffiths book equation number 3.83. Now consider my arguments.

Let lc> = Tlb>, where T is an operator, then <cl = <bl T+, where T+ is the hermitian conjugate of T. One of the property of inner product is.

<alc> = <cla>*
thus
<alTlb> = <blT+la>*

In eqn. 3.83 of Griffiths there is no conjugation when T+ operates on la>...
Does this mean

<blT+la>* = <alT+lb> ?

where * means conjugate
thanks!
 
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  • #3
Hermitian operator is just a special case of adjoints...
Sorry for the late reply ^^
 
  • #4
:uhh: Is it always true that
<blT+la>* = <alT+lb>
regardless of T being Hermitian?
 
  • #5
Of course not. Let's say you have

[tex] \langle b, T^{\dagger}a\rangle [/tex]

That's equal to

[tex] \langle (T^{\dagger})^{\dagger}b, a\rangle [/tex]

So you'd have to require that the adjoint of the adjoint should exist and moreover

[tex] T^{\dagger}b=(T^{\dagger})^{\dagger}b \ , \ \forall b\in D(T^{\dagger}) \and b\in D((T^{\dagger})^{\dagger}) [/tex]

If that happens, then you can employ Dirac's notation with bars. It's always true that an operator is included in its adjoint's adjoint, but for the adjoint it always have to be checked.

Daniel.
 
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  • #6
In the first edition of "Introduction to Quantum Mechanics" by Griffiths equation 3.83 he (Griffiths) states that a Hermitian Conjugate is an operator with the property

<alTb>=<T+alb> ... (1)

That is a Hermitian Conjugate (not necessarily Hermitian operator) is a "transformation T+ which, when applied to the first member of an inner product, gives the same result as if T itself had been applied to the second vector."

But from my previous arguments, I obtained

<alTlb>=<blT+la>* ... (2)

if i equate (1) and (2) i will have (if <alTlb> is the same as <alTb>)

<blT+a>*=<T+alb>...

notice that T+ operates on la> now.:confused:
 

1. What is the Dirac delta potential?

The Dirac delta potential, also known as the delta function potential, is a mathematical function used in quantum mechanics to describe a point-like potential that is infinitely high at a specific point and zero everywhere else. It is often used to model interactions between particles at a single point in space.

2. How is the Dirac delta potential related to the Hermitian Conjugate?

The Dirac delta potential is related to the Hermitian Conjugate through the Schrödinger equation, which is the fundamental equation in quantum mechanics. The Hermitian Conjugate of the Dirac delta potential is used to calculate the probability amplitude for a particle to be at a specific point in space.

3. What is the significance of the Hermitian Conjugate in quantum mechanics?

The Hermitian Conjugate is a mathematical operation that is used to calculate the expectation value of an operator in quantum mechanics. It is important because it allows us to calculate the probability of a particle being in a particular state, which is a key concept in quantum mechanics.

4. How is the Hermitian Conjugate of the Dirac delta potential used in real-world applications?

The Hermitian Conjugate of the Dirac delta potential is used in many real-world applications, such as modeling the interactions between particles in nuclear physics and calculating the energy levels of electrons in atoms. It is also used in signal processing and Fourier analysis to model signals that are localized in time and frequency.

5. Are there any limitations to using the Dirac delta potential and Hermitian Conjugate?

While the Dirac delta potential and Hermitian Conjugate are powerful tools in quantum mechanics, they do have some limitations. The Dirac delta potential is an idealized model and may not accurately represent real-world interactions between particles. Additionally, the Hermitian Conjugate can only be applied to certain types of operators and may not be applicable in all situations.

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