I Creation operator and Wavefunction relationship

[DeathbyGreen]

Hello,

I've noticed that some professors will jump between second quantized creation/annihilation operators and wavefunctions rather easily. For instance \Psi_k \propto c_k + ac_k^{\dagger} with "a" some constant (complex possibly). I'm fairly familiar with the second quantized notation, and from my understanding the concept of wavefunctions is essentially abandoned. The creation/annihilation operators simply add a particle into a state in the number basis. So what would be the way to "translate" some single particle wavefunciton \Psi_k into second quantization, and what would the method/reasoning be?

 

[strangerep]

I've noticed that some professors will jump between second quantized creation/annihilation operators and wavefunctions rather easily. For instance \Psi_k \propto c_k + ac_k^{\dagger} with "a" some constant (complex possibly) [...]

We'd have to see the full context that you're quoting. Sometimes a $\Psi$ like that is indeed an operator, not a state.

 

[DeathbyGreen]

I see what you're saying; in this instance the \Psi(x) is not a field operator, but a single particle wavefunction, like in the case H\Psi = \Psi E. Is it legitimate to say that \Psi(x) corresponds to a superposition of c_k's and c^{\dagger}_k's?

 

[strangerep]

... in which case writing $\Psi_k \propto c_k + ac_k^\dagger$ is nonsense (assuming the $c$'s are a/c operators).

Please give a reference, or post a link, showing the context of what you're trying to ask about.

 

[stevendaryl]

Quantum field theory can be interpreted as many-particle quantum mechanics where the number of particles is allowed to change. But what's a little confusing is the fact that the expressions \psi(\vec{r}) and \psi^\dagger(\vec{r}) don't refer to wave functions, but to operators. There is a "wave function" associated with any state with a definite number of particles, although most treatments of quantum field theory don't talk about wave functions much. But it's definitely not correct to associate a wave function with a linear combination of creation and annihilation operators; a wave function is a complex number, not an operator.

 

[vanhees71]

Well, that's why one should mark what's an operator and what's a c-number wave function, i.e., $\hat{\psi}(\vec{r})$ (field operator) vs. $\psi(\vec{r})$ (complex-valued function).

 

[Jilang]

Hello,

I've noticed that some professors will jump between second quantized creation/annihilation operators and wavefunctions rather easily. For instance \Psi_k \propto c_k + ac_k^{\dagger} with "a" some constant (complex possibly). I'm fairly familiar with the second quantized notation, and from my understanding the concept of wavefunctions is essentially abandoned. The creation/annihilation operators simply add a particle into a state in the number basis. So what would be the way to "translate" some single particle wavefunciton \Psi_k into second quantization, and what would the method/reasoning be?

I would see it as a shorthand that omits the vacuum state from the RHS. A bit sloppy but understandable.