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12b-prob1-10

12b-prob1-10 - 1 Ph 12b Homework Assignment No 1 Due 5pm...

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1 Ph 12b Homework Assignment No. 1 Due: 5pm, Thursday, 14 January 2010 1. Uncertainty principle and the quantum harmonic oscillator . For any quantum state of a single particle moving in one dimension, there is a corresponding probability density P ( x ) that governs the possible outcomes when the position of the particle is measured. P ( x ) is a nonnegative function normalized so that integraldisplay -∞ dxP ( x ) = 1 , and the expectation value ( f ) of the function f ( x ) is ( f ) = integraldisplay -∞ dxP ( x ) f ( x ) . The standard deviation Δ x of the position from its mean, defined as x ) 2 = ( ( x - ( x ) ) 2 ) , is called the position uncertainty . Similarly, another probability den- sity Q ( p ) associated with the same quantum state governs the out- comes when the momentum of the particle is measured; we may use Q ( p ) to compute expectation values of functions of p , and Δ p , the standard deviation of the momentum from its mean, is the momentum uncertainty. The position and momentum uncertainties are related by Heisenberg’s uncertainty principle, Δ x Δ p ¯ h/ 2 . a ) The energy E of a harmonic oscillator with mass m and circular frequency ω can be expressed as E = p 2 2 m + 1 2 2 x 2 . For a quantum state of the oscillator with position uncertainty Δ x and ( x ) = 0 = ( p ) , use the uncertainty principle to find a lower bound on ( E ) , expressed in terms of Δ x .
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2 b ) Now find the value of Δ x that minimizes your lower bound from part ( a ), and derive a lower bound on ( E ) that applies to any quantum state. As you will learn later in this course, the ground- state energy of a one-dimensional harmonic oscillator is E 0 = ¯ hω/ 2. Compare this value to your lower bound.
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