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Chapter 3: 3.20P (page 118)

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle (Problem 3.14), when the observable in question is x.

Short Answer

Expert verified

The energy-time uncertainty principle reduces to蟽H2蟽x2=鈩�24m2鉄�p鉄�2

Step by step solution

01

The generalized uncertainty principle for two operators.

The generalized uncertainty principle for two operators is:

蟽A2蟽B2猢�(12i鉄�[A^,B^]鉄�)2

02

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle.

From the energy-time uncertainty principle, an operator Q satisfies the equation:

ddt鉄�Q鉄�=i鈩�鉄�[H,Q]鉄�+鈭�Q鈭�t

for the position operator, Q=x, the time derivative of the position is zero, since x and t are independent variables,

ddt鉄�x鉄�=i鈩�鉄�[H,x]鉄�

Let A^=H^and B^=x^and use (2), to get:

蟽H2蟽x2猢�-鈩�2d鉄�x鉄�dt2=鈩�24m2md鉄�x鉄�dt2=鈩�24m2鉄�p鉄�2蟽H2蟽x2=鈩�24m2鉄�p鉄�2

Thus, the energy-time uncertainty principle reduces to蟽H2蟽x2=鈩�24m2鉄�p鉄�2.

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Most popular questions from this chapter

(a) Check that the eigenvalues of the hermitian operator in Example 3.1 are real. Show that the eigenfunctions (for distinct eigenvalues) are orthogonal.

(b) Do the same for the operator in Problem 3.6.

(a) Show that the set of all square-integrable functions is a vector space (refer to Section A.1 for the definition). Hint: The main problem is to show that the sum of two square-integrable functions is itself square-integrable. Use Equation 3.7. Is the set of all normalized functions a vector space?

(b) Show that the integral in Equation 3.6satisfies the conditions for an inner product (Section A.2).

(a) For what range of vis the function f(x)=x''in Hilbert space, on the interval (0.1)? Assume vis real, but not necessarily positive.

(b) For the specific case v=1/2, is f(x)in this Hilbert space? What aboutxf(x)? How about (d/dx)f(x)?

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle (Problem 3.14), when the observable in question is x.

The Hermitian conjugate (or adjoint) of an operator Q^is the operatorQ^鈥�such that

鉄�f鈭�Q^g鉄�=鉄�Q^鈥�f鈭�g鉄�鈥�(蹿辞谤补濒濒fandg).

(A Hermitian operator, then, is equal to its Hermitian conjugate:Q^=Q^鈥�)

(a)Find the Hermitian conjugates of x, i, andd/dx.

(b) Construct the Hermitian conjugate of the harmonic oscillator raising operator,a+(Equation 2.47).

(c) Show that(Q^R^)鈥�=R^鈥�Q^鈥�.
See all solutions

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