# Solving time dependent Schrodinger equation in matrix form

If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$ With Hamiltonian $H$ given by $$H=\hbar\omega \begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 2 \\ 0 & 2 & -1 \end{pmatrix}$$ With $$\psi_t(0)=(1,0,0)^T$$ So I proceeded to find the stationary states of $H$ by finding it's eigenvectors and eigenvalues. $H$ has eigenvalues and eigenvectors: $$3\hbar\omega,0,-3\hbar\omega$$ $$\psi_+=\frac{1}{3}(2,2,1)^T,\psi_0=\frac{1}{3}(2,-1,-2)^T,\psi_-=\frac{1}{3}(1,-2,2)^T$$ Respectively.

Could anyone explain to me how to go from this to a general time dependent solution, and compute probabilities of location? I have only ever encountered $\Psi=\Psi(x,y,z,t)$ before, so I am extremely confused by this matrix format.

I would be extremely grateful for any help!

-

## 1 Answer

The general solution is $$\psi(t)=\sum_k c_k e^{-itE_k/\hbar}\psi_k$$ where the $\psi_k$ form a basis of eigenvectors with corresponding eigenvalues $E_k$, and the $c_k$ are constant.

You can match arbitrary initial conditions at $t=0$ by expanding the initial state in the eigenbasis; this will determine the valued for the $c_k$.

 To get the statistical interpretation: The expectation of the Hermitian observable $A$ at time $t$ is given in the Schroedinger picture by $$\langle A\rangle_t:=\psi(t)^*A\psi(t).$$ Here it is assumed that $\psi(t)$ has norm 1. As the squared norm is preserved by the dynamics, this gives a well-defined expectation (i.e.,, the expectation of the identity matrix is 1 at all times).

-
Ok, good that's what I imagined to do, I wasn't sure. But more importantly how do you interpret this probabilistically in matrix form? – Freeman Nov 6 '12 at 20:45
@LHS: see the edited answer – Arnold Neumaier Nov 7 '12 at 8:39
Does your $c=0$ mean $c(t=0)$ ? – Nivalth Nov 7 '12 at 16:53

1
The $c_k$ are constant, and a $t$ was missing in the exponent of the first formula. $c=0$ meant $t=0$. I corrected both errors; sorry. – Arnold Neumaier Nov 8 '12 at 9:19
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 19, "mathjax_display_tex": 7, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9096881151199341, "perplexity_flag": "head"}