# Hermitian adjoint of the time derivative?

• leright
In summary, the conversation discussed the proof of the energy operator being hermitian and its relation to the Hamiltonian operator. It was also mentioned that the time derivative is not an operator in the same sense as other operators and that the Hamiltonian can take the form of a time derivative or a second spatial derivative. The conversation also included a possible proof involving the Hamiltonian operator and its components.
leright
So I had a QM test today and I needed to show that the energy operator is hermitian. This was easy to show provided that the the adjoint of d/dt is -d/dt. I know this is the case for the spatial derivative but is it the case with the time derivative? The bra-ket is an integral over x not time so the proof isn't clear to me.

The time derivative isn't an operator in the same sense as many other operators. If you have some fixed state vector $|\psi\rangle$, you cannot calculate $\partial_t|\psi\rangle$. It's not operator like that. I think, in the exam, you should have shown that the Hamiltonian operator is Hermitian.

jostpuur said:
The time derivative isn't an operator in the same sense as many other operators. If you have some fixed state vector $|\psi\rangle$, you cannot calculate $\partial_t|\psi\rangle$. It's not operator like that. I think, in the exam, you should have shown that the Hamiltonian operator is Hermitian.

wel, that's what the exam asked...it asked to show that ihbar(d/dt) is hermitian. To show that you need to assume d/dt is anti hermitian, right? oh, I see...the hamiltonian operator can take a time derivative form and a spatial derivative form...ha

leright said:
So I had a QM test today and I needed to show that the energy operator is hermitian.

leright said:
wel, that's what the exam asked...it asked to show that ihbar(d/dt) is hermitian.

So was is the energy operator, or the $i\hbar\partial_t$? The energy operator would have been the Hamiltonian $H$, that depends on the system.

Can I prove this way?

$$<\phi|H|\phi> = <\phi|E|\phi> = <\phi|E^*|\phi> = <\phi|H^+|\phi>$$

So

$$H=H^+$$

jostpuur said:
So was is the energy operator, or the $i\hbar\partial_t$? The energy operator would have been the Hamiltonian $H$, that depends on the system.

that is the energy operator. the hamiltonian can take the form of a time derivative or a second spatial derivative...I think.

leright said:
that is the energy operator. the hamiltonian can take the form of a time derivative or a second spatial derivative...I think.

When $|\psi(t)\rangle$, a state vector that is a function of time, satisfies the Schrödinger's equation

$$i\hbar\partial_t|\psi(t)\rangle = H|\psi(t)\rangle,$$

then, by definition, the operators $i\hbar\partial_t$ and $H$ will have the same effect on any such vector, but in general these are different operators. They must be of course, since if they were the same, the SE would have no content about the time evolution of the state!

In other words, if $|\xi(t)\rangle$ is some arbitrary parametrization of a path in the state space, then you can have

$$i\hbar\partial_t|\xi(t)\rangle \neq H|\xi(t)\rangle.$$

So, they are different operators, and it is correct to write

$$i\hbar\partial_t \neq H.$$

Last edited:
Sorry to revive an old thread, but would this proof work?

We know that

$$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)$$

Now, clearly, $$V(x)$$ is Hermitian. In addition, we can ignore all multiplicative constants in the problem (a multiple of a Hermitian operator is still Hermitian). But we know that $$d/dx$$ is anti-Hermitian, so this means that $$(d^2 / d x^2)^\dagger = d^2/dx^2$$, implying that the Hamiltonian operator is Hermitian.

## What is the definition of the Hermitian adjoint of the time derivative?

The Hermitian adjoint of the time derivative is a mathematical operation that involves taking the complex conjugate of a function's time derivative and then flipping the sign of the imaginary part. This operation is commonly used in quantum mechanics and other areas of physics to describe the evolution of a system over time.

## Why is the Hermitian adjoint of the time derivative important?

The Hermitian adjoint of the time derivative allows us to describe the time evolution of quantum systems in a way that is consistent with the principles of quantum mechanics. It also has practical applications in fields such as signal processing and control theory.

## How is the Hermitian adjoint of the time derivative related to the Schrödinger equation?

The Schrödinger equation, which describes the time evolution of a quantum system, can be written in terms of the Hermitian adjoint of the time derivative. This allows us to use the mathematical machinery of the Hermitian adjoint to solve the Schrödinger equation and make predictions about the behavior of quantum systems.

## Can the Hermitian adjoint of the time derivative be applied to non-quantum systems?

Yes, the Hermitian adjoint of the time derivative can be applied to any system that can be described using complex numbers. It is commonly used in fields such as signal processing, control theory, and quantum mechanics, but it has applications in other areas of science and engineering as well.

## What are some other names for the Hermitian adjoint of the time derivative?

The Hermitian adjoint of the time derivative is also known as the adjoint operator, the adjoint of the time derivative, or the adjoint of the derivative. In some contexts, it may also be referred to as the Hermitian conjugate or the adjoint of the operator.

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