Schrodinger Equation Notation: Vector or Scalar?

[eep]

Hi,
I've seen the Schrödinger equation written in the following form:

$$i\hbar\frac{\partial\Psi}{{\partial}t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi$$

where

$$\nabla^{2} = \frac{\partial^{2}}{{\partial}x^2} + \frac{\partial^2}{{\partial}y^2} + \frac{\partial^2}{{\partial}z^2}$$

Now, is $\nabla^2\Psi$ a vector or a scalar? In this notation, I would say it's a vector. You have $\nabla^2$ acting on each of the components of $\Psi$. However the book seems to say that $\nabla^2\Psi$ is a scalar. Shouldn't the notation then be $\nabla^2\cdot\Psi$? That is, shouldn't it be a dot product? I'm rather confused...

 

[nrqed]

Hi,
I've seen the Schrödinger equation written in the following form:

$$i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi$$

where

$$\nabla^2 = \frac{{\partial}^2}{{\partial}x^2} + \frac{partial^2}{\partialy^2} + \frac{\partial^2}{\partialz^2}$$

Now, is $\nabla^2\Psi$ a vector or a scalar? In this notation, I would say it's a vector. You have $\nabla^2$ acting on each of the components of $\Psi$. However the book seems to say that $\nabla^2\Psi$ is a scalar. Shouldn't the notation then be $\nabla^2\cdot\Psi$? That is, shouldn't it be a dot product? I'm rather confused...

You seem to thing of $\Psi$ as a 3-D vector with i,j,k component. That`s not the case. It is a scalar function.

Pat

 

[eep]

Can't we represent $\Psi$ as a vector in hilbert space?

 

[Hurkyl]

Yes -- and the vectors in that Hilbert space are the scalar functions. They're not geometric vectors.

(There's a reason you learned about vector spaces other than the n-tuples in your linear algebra course!)

 

[nrqed]

Can't we represent $\Psi$ as a vector in hilbert space?

(I *thought* we would be getting to this point but I did not want to muddle the waters too fast). You are perfectly right. Except that this use of the word vector has nothing to do with the usual use of vector as meaning soemthing of the form $A_x {\vec i} + A_y {\vec j} + A_z {\vec k}$. The solutions of Schrödinger`s equations form a vector space in the more general mathematical sense, but they are each scalar functions.

Pat

 

[eep]

Ah, of course. I was thinking that when I posted but didn't want to jump the gun either. So the Schrödinger equation acts on $\Psi$ which is a function of x,y,z,t and whose range is scalars. Each $\Psi$ lives in Hilbert space and can be represented by a vector, but this has nothing to do with the way the Schrödinger equation acts on $\Psi$, right?

 

[reilly]

DEL**2 is a scalar (under rotations). This follows from a standard vector calculus convention, that DEL or GRADIENT is a vector with the following components (d/dx,d/dy/d/dz). Discussed in thousands of textbooks,
Regards,
Reilly Atkinson

 

[gulsen]

I have another question about notation in QM.
If $$< \Psi | \hat{H} \Psi>$$ is the same thing as $$< \Psi | \hat{H} | \Psi>$$, why do they invent the extra | between?

 

[ZapperZ]

I have another question about notation in QM.
If $$< \Psi | \hat{H} \Psi>$$ is the same thing as $$< \Psi | \hat{H} | \Psi>$$, why do they invent the extra | between?

Because that is only valid for a hermitian operator. If you replace that with a non-hermition operator in a purely mathematical exercise, then these two are no longer identical.

Zz.

 

[Perturbation]

Hi,
I've seen the Schrödinger equation written in the following form:

$$i\hbar\frac{\partial\Psi}{{\partial}t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi$$

where

$$\nabla^{2} = \frac{\partial^{2}}{{\partial}x^2} + \frac{\partial^2}{{\partial}y^2} + \frac{\partial^2}{{\partial}z^2}$$

Now, is $\nabla^2\Psi$ a vector or a scalar? In this notation, I would say it's a vector. You have $\nabla^2$ acting on each of the components of $\Psi$. However the book seems to say that $\nabla^2\Psi$ is a scalar. Shouldn't the notation then be $\nabla^2\cdot\Psi$? That is, shouldn't it be a dot product? I'm rather confused...

For one, it's in coordinate representation, so the wave-function is a scalar given by $\langle x|\psi\rangle$. Here the bra and ket vectors themselves aren't vectors in the usual sense (an n-tuple), they're members of a Hilbert space, an infinite dimensional linear vector space over $\mathbb{C}$, where the members of the space are functions (which obviously satisfy the vector space axioms).

I have another question about notation in QM.
If $$< \Psi | \hat{H} \Psi>$$ is the same thing as $$< \Psi | \hat{H} | \Psi>$$, why do they invent the extra | between?

Because H is self adjoint.

 

[Physics Monkey]

I have another question about notation in QM.
If $$< \Psi | \hat{H} \Psi>$$ is the same thing as $$< \Psi | \hat{H} | \Psi>$$, why do they invent the extra | between?

It is really a notational choice and not much more (why keep the extra bar). You can define the ket $$| H \psi \rangle$$ as $$\hat{H} | \psi \rangle$$. What you cannot do is write $$\langle \psi | \hat{H} | \psi \rangle = \langle H \psi | \psi \rangle$$ unless $$\hat{H}$$ is self adjoint. In fact, $$\langle H^\dag \psi | \psi \rangle = \langle \psi | H \psi \rangle = \langle \psi | \hat{H} | \psi \rangle$$ where as $$\langle H \psi | \psi \rangle = \langle \psi | H^\dag \psi \rangle = \langle \psi | \hat{H}^\dag | \psi \rangle$$