How would you know if a collection of objects is a vector space?
Let's take vector space <math>V</math>, with vectors <math>u</math>, <math>v</math>, <math>w</math> in it.
- <math>u + v</math> belongs in <math>V</math>
- <math>uv</math> belongs in <math>V</math>
- <math>(u + v) + w = u + (v + w)</math>
- <math>u + 0 = u</math>
- <math>u + (-u) = 0</math>
- <math>u + v = v + u</math>
- <math>k(u + v) = ku + kv</math> where <math>k</math> is a scalar
- <math>(a + b)u = au + bu</math> where <math>a</math> and <math>b</math> are scalars
- <math>(ab)u = a(bu)</math>
- <math>1u = u</math>
Wha is the equivalent of a projection in function space?
The dot product
, also known as the inner product. An inner product between two functions is written below.
<center>
<br>
<math><u
|\Psi>=\int u^* \Psi dx</math>
<br>
</center>
Consider the geometric interpretation in two dimensions for simplicity. Consider two vectors <math>\Psi</math> and <math>u</math>. The inner product results in a number that could be a complex scalar that tells the projection of one vector on the other. The projection is independent of the basis the vector is in.
What are the special properties of hermitian operators?
These important characterisitics are associated with Hermitian operators
- All the eigenvalues are real
- Eigenvectors belonging to different eigenvalues are orthogonal to one another
- A Hermitian operator operating on a space of <math>N</math> dimentions will have <math>N</math> linearly independent eigenvectors
- Operator <math>A</math> is Hermitian if <math>A</math> adjoint equals <math>A</math>
<center>
<br>
<math>\hat A^t = \hat A</math>
<br>
<math>\hat A_
^t = \hat A_
^*</math>
<br>
<math>\hat A_
= \hat A_
^*</math>
<br>
<math><\Phi_1 \mid \hat A \Phi_2> = <\hat A \Phi_1 \mid \Phi_2></math>
</center>
<br>
How do we represent in QM the basic position (X) and momentum (P) observables?
<center>
<br>
<math> \hat P = \mbox
</math>
<br>
<math> \hat P = \frac
\vec \nabla </math>
<br>
<math> \hat R = \mbox
</math>
<br>
<math> \hat R = \hat r </math>
</center>
<br>
How do we construct QM observables (free particle Hamiltonian)?
Energy:
<center>
<br>
<math> \hat H (\hat x, \hat p)=i\hbar\frac
=(\frac
+V(\hat x)) </math>
<br>
<math> \frac
=\frac {-\hbar^2}
\frac
</math>
<br>
</center>
General soluion to Schr��dinger's equation:
<center>
<br>
<math> \Psi(x,t)= \sum_
c_E u_E \mathrm
^{-\mathrm
Et/ \hbar} </math>
<br>
</center>
where <math> c_E </math> are the eigenfunctions of the Hamiltonian that has eigenvalues of <math>E</math>.
How do we find the time evolution of a state?
Below is the time-independent and time-dependent Schr��dinger's equation. To find the time evolution, we use the time-dependent equation.
<center>
<br>
<math>\mbox
</math>
<br>
<math>\hat H \Psi = E \Psi</math>
<br>
<math>\mbox
</math>
<br>
<math>\hat H \Psi = i \hbar \frac
</math>
<br>
</center>
For Conservative Systems
If we have a conservative system (system where the total energy is conserved), we can use a solution in the form of <math>\Psi = \phi \xi (t)</math> where <math>\xi (t)</math> is the time-dependent part. Use <math> \phi </math> with the time-independent Schr��dinger's equation, and <math>\xi (t)</math> with the time-dependent Schr��dinger's equation.
Specifically, the solution looks like <math>u_E e^{-i \frac
t}</math> where <math>u_E </math> is an eigenfunction of the Hamiltonian with eigenvalues <math>E</math>. The superposition of solutions are also solutions. The general solution of Schr��dinger's equation for conservative systems is below.
<center>
<br>
<math> \Psi(x,t)= \sum_
c_E u_E \mathrm
^{-\mathrm
Et/ \hbar} </math>
<br>
</center>
If the spatial wavefunction at any given time is known, it can be predicted at any other time. The key is to find the energy eigenfunctions. The form of the eigenfunctions is below.
<center>
<br>
<math>u_n = \begin
n \mbox
\sqrt{\frac
{d}} \cos k_n x = \sqrt{\frac
{d}} \cos \frac
n \mbox
\sqrt{\frac
{d}} \sin k_n x = \sqrt{\frac
{d}} \sin \frac
\end
</math>
<br>
<math>k_n = \sqrt{\frac
{\hbar^2}} = \frac
</math>
</center>