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Quantum Mechanics Problem: Manipulating Bra-Ket Notation, Assignments of Chemistry

A problem from a university-level quantum mechanics course that involves manipulating bra-ket notation. Students are required to normalize and find the probability of finding a particle in the 'up' or 'down' state, find a ket orthogonal to a given ket, express the 'up' and 'down' kets in terms of the new kets, and calculate the average rate of clicks from a measurement device based on the initial state and the measurement outcomes.

Typology: Assignments

Pre 2010

Uploaded on 08/18/2009

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koofers-user-3gv 🇺🇸

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Chem 212
Problem 1. (Due Thursday, September 4, 2008)
This problem is meant to give you practice manipulating bras and kets.
A two-state system can be represented by the kets |u> and |d> depending
whether the particle is in the “up” or “down” state. Assume that the |u> and |d>
kets are normalized and orthogonal.
(a) If the ket for a prepared state of the particle is known to be
|
α
+>= 1
2|u> + 3
2|d>
then show that |α+> is normalized, and find the probability that the particle is
(i) up or (ii) down.
(b) Find a ket that is orthogonal to |α+> and normalize it. Call it |α>.
(c) Express the kets |u> and |d> in terms of the kets |α+> and |α>.
(d) If a ket is known to be in the |u> state, then what is the probability that
it is in the |α+> state? The |α> state?
(e) Suppose that the particle is known to be in |α+> initially, and is passed
through a measuring device that measures only |u> or |d>. Particles
that are measured as “up” are deflected up, and those that are
measured “down” are deflected down. Another measurement device
follows which measures whether the state is |α+> or |α>. The “+”
response triggers and audible detector as a “click”. If a beam of 1000
particles per second is sent through the first detector, what is the
average rate of clicks?

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Chem 212 Problem 1. (Due Thursday, September 4, 2008) This problem is meant to give you practice manipulating bras and kets. A two-state system can be represented by the kets |u> and |d> depending whether the particle is in the “up” or “down” state. Assume that the |u> and |d> kets are normalized and orthogonal. (a) If the ket for a prepared state of the particle is known to be €

| α+ >= 12 | u > + 23 | d >

then show that |α+> is normalized, and find the probability that the particle is (i) up or (ii) down. (b) Find a ket that is orthogonal to |α+> and normalize it. Call it |α−>. (c) Express the kets |u> and |d> in terms of the kets |α+> and |α−>. (d) If a ket is known to be in the |u> state, then what is the probability that it is in the |α+> state? The |α−> state? (e) Suppose that the particle is known to be in |α+> initially, and is passed through a measuring device that measures only |u> or |d>. Particles that are measured as “up” are deflected up, and those that are measured “down” are deflected down. Another measurement device follows which measures whether the state is |α+> or |α−>. The “+” response triggers and audible detector as a “click”. If a beam of 1000 particles per second is sent through the first detector, what is the average rate of clicks?