The eigenvalues and eigenfunctions of a quantum mechanical operator A are denoted by an and Ψn respectively. If f(x) denotes a function that can be expanded in the powers of x, show that:
To solve this numerically, let’s first understand the problem analytically. ### Step 1: Understanding the Operator Equation Given: \[ A \Psi_n = a_n \Psi_n \] where \( A \) is a quantum mechanical operator, and \( a_n \) and \( \Psi_n \) are its eigenvalues and eigenfunctions, respectively. If \( f(x) ______ _____ _____ _______ ____ _________ _____.
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I) कारण सहित बताएं कि निम्नलिखित अभिक्रियाओं में से कौन सी संभव है?
i)
ii)
iii)
एक क्वांटम यांत्रिकीय संकारक A के आइगेन मान और आइगेन फलन क्रमशः anऔर Ψn है। यदि f(x) एक ऐसा फलन है जिसका प्रसार x की घात के पदों में हो सकता है तो सिद्ध करें कि :
हाइड्रोजन परमाणु के फलन तरंगों (i) Ψ210 और (ii) Ψ300 के मान लिखें। हाइड्रोजन परमाणु की मूल अवस्था के लिए r का प्रत्याशा मान प्राप्त करें।
निम्नलिखित कणों को बेरिऑन, मीसॉन और लेप्टॉन में वर्गीकृत कीजिएः
Neutral hydrogen atoms are moving along the axis of an evacuated tube with a speed of 2.0 × 106 ms-1 . A spectrometer is arranged to receive light emitted by these atoms in the direction of their forward motion. This light, if emitted from resting hydrogen atoms, would have a measured wavelength of 486.13 nm. Calculate the expected wavelength for light emitted from the approaching atoms, using the relativistic formula.
A spacecraft travels past Earth and Mars in a straight line at a speed v = 0.8c at an instant when Earth and Mars are 2.4 × 1011 m apart; the distance being measured in the fixed frame of reference in which Earth and Mars are at rest. How far apart are Earth and Mars in the frame of the spacecraft? In the frame of the spacecraft, how much time elapses between the spacecraft crossing Earth and the spacecraft reaching Mars?
An electron microscope uses an electron beam of energy 1.0 keV. Can this microscope be used to obtain the image of an individual atom? (The size of an atom ≈ 10-10 m.)