What is the Complex Conjugate of a Hermitian Integral in QM?

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In summary: No, Atkin's Physical Chemistry. QM is interesting though. Might take a graduate level elective if I pass this course with a decent grade.
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Mayhem
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My QM textbook defines Hermiticity as $$\int f^*\hat{\Omega}g dx = \left \{ \int g^*\hat{\Omega}f dx \right\}^*$$ where f and g are any two wave functions, and * denotes the complex conjugate.

I am having a little trouble interpreting the complex conjugate of the RHS integral. Usually the complex conjugate of a function is defined as ## \psi^* = (f+gi)^* = f-gi ## (here f and g are not necessarily related to the above definition). Can I make a similar decomposition of the integral and is this even useful?
 
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Mayhem said:
My QM textbook defines Hermiticity as $$\int f^*\hat{\Omega}g dx = \left \{ \int g^*\hat{\Omega}f dx \right\}^*$$ where f and g are any two wave functions, and * denotes the complex conjugate.

I am having a little trouble interpreting the complex conjugate of the RHS integral. Usually the complex conjugate of a function is defined as ## \psi^* = (f+gi)^* = f-gi ## (here f and g are not necessarily related to the above definition). Can I make a similar decomposition of the integral and is this even useful?
Your textbook omits to show that those integrals are definite integrals and hence complex numbers, not functions of ##x##.
 
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PeroK said:
Your textbook omits to show that those integrals are definite integrals and hence complex numbers, not functions of ##x##.
Ah, that makes sense. Actually they do show an example where there are limits. So this means that it is necessary to evaluate the integral in order to extract an explicit form of the complex conjugate?
 
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Mayhem said:
Ah, that makes sense. Actually they do show an example where there are limits. So this means that it is necessary to evaluate the integral in order to extract an explicit form of the complex conjugate?
You don't necessarily have to evaluate the integral.
 
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Mayhem said:
My QM textbook defines Hermiticity as $$\int f^*\hat{\Omega}g dx = \left \{ \int g^*\hat{\Omega}f dx \right\}^*$$ where f and g are any two wave functions, and * denotes the complex conjugate.

I am having a little trouble interpreting the complex conjugate of the RHS integral. Usually the complex conjugate of a function is defined as ## \psi^* = (f+gi)^* = f-gi ## (here f and g are not necessarily related to the above definition). Can I make a similar decomposition of the integral and is this even useful?
You have an inner product defined by a bilinear, self-adjoint operator ##\hat{\Omega}=\hat{\Omega}^*=\overline{\hat{\Omega}}##
$$
\langle \overline{f}\, , \,g \rangle_\hat{\Omega}= \int f^*\hat{\Omega}g dx = \left \{ \int g^*\hat{\Omega}f dx \right\}^*=\overline{\langle \overline{g},f \rangle}_\hat{\Omega}
$$
 
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fresh_42 said:
You have an inner product defined by a bilinear, self-adjoint operator ##\hat{\Omega}=\hat{\Omega}^*=\overline{\hat{\Omega}}##
$$
\langle \overline{f}\, , \,g \rangle_\hat{\Omega}= \int f^*\hat{\Omega}g dx = \left \{ \int g^*\hat{\Omega}f dx \right\}^*=\overline{\langle \overline{g},f \rangle}_\hat{\Omega}
$$
That doesn't look right. That's the definition of Hermicity of ##\Omega##, using the standard inner product on the space of square-integrable functions.

That's what the OP gets for posting QM in a maths forum!
 
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Let me guess: The textbook is Griffiths...?
 
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vanhees71 said:
Let me guess: The textbook is Griffiths...?
No, Atkin's Physical Chemistry. QM is interesting though. Might take a graduate level elective if I pass this course with a decent grade.
 
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FAQ: What is the Complex Conjugate of a Hermitian Integral in QM?

What is a Hermitian integral in quantum mechanics?

A Hermitian integral in quantum mechanics is a type of mathematical operation used to calculate the expectation value of an observable in a quantum system. It is a complex integral that takes into account both the wave function and its complex conjugate, and is an important tool in understanding the behavior of quantum systems.

What does it mean for an integral to be Hermitian?

In quantum mechanics, a Hermitian integral is one that satisfies the condition of Hermiticity, which means that the integral is equal to its own complex conjugate. This property is important because it ensures that the expectation value calculated using the integral is a real number, which is necessary for physical observables in quantum mechanics.

What is the complex conjugate of a Hermitian integral?

The complex conjugate of a Hermitian integral is the integral with its complex terms reversed in sign. In other words, the complex conjugate of an integral is obtained by replacing all the imaginary numbers with their negative counterparts. This is necessary to satisfy the condition of Hermiticity and ensure that the integral is equal to its own complex conjugate.

How is the complex conjugate used in calculating expectation values?

In quantum mechanics, the complex conjugate is used in conjunction with the wave function to calculate the expectation value of an observable. The complex conjugate is multiplied by the wave function in the integral, and the resulting expression is integrated over the entire space. This gives the expectation value, which is a real number representing the average value of the observable in the quantum system.

Why is the complex conjugate important in quantum mechanics?

The complex conjugate is important in quantum mechanics because it allows for the calculation of real-valued expectation values, which are necessary for physical observables. Additionally, the complex conjugate is used in many other mathematical operations and equations in quantum mechanics, making it a fundamental concept in the field.

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