Quantum Qual Problems - Revision history

Oborn at 21:13, 17 August 2007

2007-08-17T21:13:31Z

2007-08-17T21:10:13Z

2007-08-17T21:09:27Z

2007-08-17T21:07:49Z

2007-08-17T21:07:02Z

2007-08-17T21:06:37Z

2007-08-17T21:06:25Z

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2007-08-17T21:02:42Z

← Older revision		Revision as of 21:13, 17 August 2007
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−	b.) Assume that <math>w_1 - w_2 >> \|v\|</math>	+	b.) Assume that <math>w_1 - w_2 >> \|v\|</math>. Use time independent perturbation theory to compute the first and second order corrections to the energy levels of <math>H_0</math> when <math>H_1</math> is treated as a perturbation.
		+
		+	c.) Let <math>w=w_1=w_2</math> in <math>H_0</math> above. Assuming the system is initially in the eigenstate
		+	<math>\begin{pmatrix}1 \\ 0\end{pmatrix}</math> of <math>H_0</math>, and the interaction <math>H_1</math> is turned on for a finite time interval <math>T</math>. Find the probabitlity that the system will be found in the state <math>H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}</math> at the end of that time interval.

@@ Line 19: / Line 19: @@
 <math>H_1= \begin{pmatrix} 0 & v \\ v & 0\end{pmatrix}</math>
 where v is real.
-Find the exact values of the energies of the new Hamitonlian, <math>H=H_0 + H_1</math>

@@ Line 12: / Line 12: @@
 <math>\begin{pmatrix}1 \\ 0\end{pmatrix}</math>
 and
-<math>\begin{pmatrix} 0 \\ 1\end{pmatrix}</math>
+<math>\begin{pmatrix} 0 \\ 1\end{pmatrix}</math>.
-the Hamiltonian can be written as
+The Hamiltonian can be written as
 <math>H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}</math>

@@ Line 13: / Line 13: @@
 and
 <math>\begin{pmatrix} 0 \\ 1\end{pmatrix}</math>
 the Hamiltonian can be written as

@@ Line 10: / Line 10: @@
 .) A system has two energy eigenstate with eigenvalues <math>w_1</math> and <math>w_2</math>.  Assume that <math>w_1 > w_2</math>.  Representing the enegy eigenstate by
-<math>\begin{pmatrix}
+<math>\begin{pmatrix}1 \\ 0\end{pmatrix}</math>
-\\
+and
-\end{pmatrix}</math>
+<math>\begin{pmatrix} 0 \\ 1\end{pmatrix}</math>
 the Hamiltonian can be written as

← Older revision		Revision as of 21:06, 17 August 2007
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	2.) A system has two energy eigenstate with eigenvalues <math>w_1</math> and <math>w_2</math>. Assume that <math>w_1 > w_2</math>. Representing the enegy eigenstate by		2.) A system has two energy eigenstate with eigenvalues <math>w_1</math> and <math>w_2</math>. Assume that <math>w_1 > w_2</math>. Representing the enegy eigenstate by
		+	<math>A = \begin{pmatrix}
		+	1 \\
		+	0\end{pmatrix}</math>
		+


	the Hamiltonian can be written as		the Hamiltonian can be written as

@@ Line 11: / Line 11: @@
 .) A system has two energy eigenstate with eigenvalues <math>w_1</math> and <math>w_2</math>.  Assume that <math>w_1 > w_2</math>.  Representing the enegy eigenstate by
-<math>\left ( \begin{array} {c} 1 \\ 0 \end{array} \right ) </math>
+<math> \begin{array} {c} 1 \\ 0 \end{array}  </math>
 the Hamiltonian can be written as